Simplify the expressions using exponent rules.
-
$b^4 \cdot b^3$ Solution: When multiplying expressions with the same base, we add the exponents. So,
$b^4 \cdot b^3 = b^{4+3} = b^7$ . -
$c^5 \cdot c^2 \cdot c^2$ Solution: Similarly, for this expression, we add all the exponents:
$c^5 \cdot c^2 \cdot c^2 = c^{5+2+2} = c^9$ . -
$a^{-4} \cdot a^{-3}$ Solution: The rule of adding exponents applies here as well. Negative exponents indicate that the base is on the denominator of a fraction. However, for this problem, we just add the exponents:
$a^{-4} \cdot a^{-3} = a^{-4-3} = a^{-7}$ . -
$x^5 \cdot x^{-4} \cdot x$ Solution: In this case, we add the exponents, and remember that
$x$ is the same as$x^1$ . So,$x^5 \cdot x^{-4} \cdot x = x^{5+(-4)+1} = x^2$ . -
$(2x)^2 \cdot (4y)^2$ Solution: When raising a power to a power, we multiply the exponents. Also, remember that both the coefficient and the variable are raised to the power. So,
$(2x)^2 \cdot (4y)^2 = (2^2x^2)(4^2y^2) = 4x^2 \cdot 16y^2 = 64x^2y^2$ . -
$-2gh \cdot (g^3h^5)$ Solution: In this case, we multiply the like terms. So,
$-2gh \cdot (g^3h^5) = -2g^1h^1 \cdot g^3h^5 = -2g^{1+3}h^{1+5} = -2g^4h^6$ . -
$10x^2y^3 \cdot (10xy^8)$ Solution: Here, we multiply the coefficients and add the exponents of like terms. So,
$10x^2y^3 \cdot (10xy^8) = 10 \cdot 10 \cdot x^{2+1}y^{3+8} = 100x^3y^{11}$ . -
$\frac{24wz^2}{3w^3z^5}$ Solution: For this expression, we simplify the coefficients by division and subtract the exponents of like terms (numerator - denominator). So,
$\frac{24wz^2}{3w^3z^5} = \frac{24}{3} \cdot w^{1-3}z^{2-5} = 8w^{-2}z^{-3}$ . This can also be written as$\frac{8}{w^2z^3}$ if you prefer not to have negative exponents. -
$c^{\frac{2}{3}}$ Solution: The expression
$c^{\frac{2}{3}}$ can be rewritten in radical notation as$\sqrt[3]{c^2}$ . This is because the denominator of the fraction in the exponent represents the root (in this case, the cube root), and the numerator represents the power (in this case, squared). -
$x^{\frac{1}{6}}$
Solution: Similarly, the expression
$\sqrt{16x^{13}}$
Solution: The square root of 16 is 4. For
$\sqrt[3]{27a^6b^2}$
Solution: The cube root of 27 is 3. For
$\sqrt{(3c)^9}$
Solution: When taking the square root of a term raised to an odd power, we can take out half of the power (rounded down) and leave the rest inside the square root. The square root of
$\sqrt{2} + \sqrt{50} - 4\sqrt{8}$
Solution: We can simplify each term first.
$4\left(2\sqrt{9} - 5\right)$
Solution: We can simplify inside the parentheses first.
$(1+2\sqrt{7})(5 + \sqrt{7})$
Solution: We can use the distributive property (also known as the FOIL method) to expand this expression.
First, we multiply the first terms in each binomial:
Next, we multiply the outer terms:
Then, we multiply the inner terms:
Finally, we multiply the last terms:
Adding these results together, we get
- Find the area of the triangle given the following characteristics:
$b = 10x$ ,$h = 4x^2$ ,$A = \frac{1}{2}bh$
Solution: We can substitute the given values of
Substituting
So, the area of the triangle is
$\sqrt{-144}$
Solution: The square root of a negative number is an imaginary number. The square root of -144 is equal to the square root of 144 times the square root of -1. The square root of 144 is 12, and the square root of -1 is denoted as
$i^3$
Solution: The power of
$(6i - 8) + (-3i + 2)$
Solution: We can combine like terms to simplify this expression.
$4i(i+2)$
Solution: We can distribute
$\frac{12}{3-i}$
Solution: To simplify a complex fraction, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of
$\sqrt{-2} \cdot \sqrt{-9}$
Solution: The square root of a negative number is an imaginary number. The square root of -2 is
$f(x) = -\sqrt{x} + 3 + 2$
Solution:
Transformations:
- The negative sign in front of the square root function reflects the graph over the x-axis.
- The "+3" inside the square root function shifts the graph 3 units to the left.
- The "+2" outside the square root function shifts the graph 2 units up.
To graph this function, start with the graph of
$f(x) = 2\sqrt{x} - 1 - 4$
Solution:
Transformations:
- The "2" in front of the square root function vertically stretches the graph by a factor of 2.
- The "-1" inside the square root function shifts the graph 1 unit to the right.
- The "-4" outside the square root function shifts the graph 4 units down.
To graph this function, start with the graph of
Note: For both functions, the transformations are applied in the order of reflection, horizontal shift, and then vertical shift.
$\sqrt{x} + 3 = 5$
Solution: To solve this equation, we first isolate the square root term by subtracting 3 from both sides of the equation:
$3\sqrt{x} - 2 = 7$
Solution: First, we isolate the square root term by adding 2 to both sides of the equation:
[example video][https://youtu.be/H46QVUdbBn0] [practice problem][https://www.ixl.com/math/algebra-2/graph-exponential-functions] [caracteristics of function graph][https://youtu.be/woVwBmwhA2Y]
Exponential Form & Logarithmic Form
Identify the characteristics and graph the following functions.
Characteristics:
- Type: Growth
- Domain: All real numbers
- Range: All real numbers greater than or equal to 1
Graph:
To graph the function, we can plot a few points to get an idea of the shape of the graph. Let's choose some x-values and calculate the corresponding y-values:
For
For
For
For
For
Now, let's plot these points on a graph and connect them to get the shape of the exponential function
The graph of the function
Identify the characteristics and graph the following functions.
Characteristics:
- Vertical Asymptote (VA):
$x = 0$ - x-intercept (X-Int):
$x = 1$ - Domain:
$x > 0$ - Range: All real numbers
Graph:
To graph the function, we can plot some points and draw the graph:
-
For
$x = \frac{1}{8}$ :$f\left(\frac{1}{8}\right) = \log_{\frac{1}{2}}\left(\frac{1}{8}\right) = -3$ -
For
$x = \frac{1}{4}$ :$f\left(\frac{1}{4}\right) = \log_{\frac{1}{2}}\left(\frac{1}{4}\right) = -2$ -
For
$x = \frac{1}{2}$ :$f\left(\frac{1}{2}\right) = \log_{\frac{1}{2}}\left(\frac{1}{2}\right) = -1$ -
For
$x = 1$ :$f(1) = \log_{\frac{1}{2}}(1) = 0$ -
For
$x = 2$ :$f(2) = \log_{\frac{1}{2}}(2) = 1$ -
For
$x = 4$ :$f(4) = \log_{\frac{1}{2}}(4) = 2$ -
For
$x = 8$ :$f(8) = \log_{\frac{1}{2}}(8) = 3$
By plotting these points and considering the vertical asymptote at
Remember to label the axes and indicate any asymptotes or intercepts on the graph.
Write an exponential function for the graph that passes through the given points. Then determine if the function is growth or decay.
Given Points:
- Point 1: (0, 3)
- Point 2: (2, 192)
To write an exponential function that passes through the given points, we'll use the general form of an exponential function:
where
- Finding the value of
$a$ : Using the point (0, 3), we have:
Since any number raised to the power of 0 is equal to 1, we can simplify to:
Therefore, the value of
- Finding the value of
$b$ : Using the point (2, 192), we have:
Dividing both sides by 3:
Taking the square root of both sides:
Since exponential functions can't have a negative base, we'll take the positive value:
Therefore, the exponential function that passes through the given points is:
To determine whether the function is growth or decay, we observe that the base
Write an exponential function for the graph that passes through the given points. Then determine if the function is growth or decay.
Given Points:
- Point 1: (0, 20)
- Point 2: (2, 5)
To write an exponential function that passes through the given points, we'll use the general form of an exponential function:
where
- Finding the value of
$a$ : Using the point (0, 20), we have:
Since any number raised to the power of 0 is equal to 1, we can simplify to:
Therefore, the value of
- Finding the value of
$b$ : Using the point (2, 5), we have:
Dividing both sides by 20:
Taking the square root of both sides:
Since exponential functions can't have a negative base, we'll take the positive value:
Therefore, the exponential function that passes through the given points is:
To determine whether the function is growth or decay, we observe that the base
Write an exponential function for the graph that passes through the given points. Then determine if the function is growth or decay.
Given Points:
- Point 1: (0, 1)
- Point 2: (3, 216)
To write an exponential function that passes through the given points, we'll use the general form of an exponential function:
where
- Finding the value of
$a$ : Using the point (0, 1), we have:
Since any number raised to the power of 0 is equal to 1, we can simplify to:
Therefore, the value of
- Finding the value of
$b$ : Using the point (3, 216), we have:
Simplifying:
Taking the cube root of both sides:
Therefore, the exponential function that passes through the given points is:
To determine whether the function is growth or decay, we observe that the base
The equation
First, we need to express both sides of the equation in terms of the same base. We know that
Simplifying the right side of the equation gives us:
Since the bases are the same, we can equate the exponents:
Solving this equation for
Here is a plot of the equation, which confirms that
For more details, you can check the Wolfram|Alpha result.
Please note that the equation also has complex solutions, but since we are dealing with real numbers, we only consider the real solution.
The equation
First, we need to express both sides of the equation in terms of the same base. We know that
Simplifying the right side of the equation gives us:
Since the bases are the same, we can equate the exponents:
Solving this equation for
Here is a plot of the equation, which confirms that
For more details, you can check the Wolfram|Alpha result.
Please note that the equation also has complex solutions, but since we are dealing with real numbers, we only consider the real solution.
The equation
The base
This is read as "log base
The equation
The base
This is read as "log base
The equation
The base of the logarithm
This is read as "8 raised to the power of
The equation
The base of the logarithm
This is read as "$b$ raised to the power of
The logarithm
The answer to this is
Here is a visual representation of the number
For more details, you can check the Wolfram|Alpha result.
The logarithm
The answer to this is
Here is a number line representation of the number
For more details, you can check the Wolfram|Alpha result.
The logarithm
The answer to this is
Here is a visual representation of the number
For more details, you can check the Wolfram|Alpha result.
The logarithm
The answer to this is
Here is a number line representation of the number
For more details, you can check the Wolfram|Alpha result.
- Rational Expressions: Introduction to Rational Expressions by Khan Academy
- Multiplying and Dividing Rational Expressions: Multiplying and Dividing Rational Expressions by Mathispower4u
- Adding and Subtracting Rational Expressions: Adding and Subtracting Rational Expressions by Khan Academy
- Solving Rational Equations: Solving Rational Equations by The Organic Chemistry Tutor
- Graphing Rational Functions: How to graph a rational function using 6 steps by Brian McLogan
Here are the steps to multiply the rational expressions
-
Identify the Numerator and Denominator of Each Fraction: The first fraction has
$m^2+8m+15$ as its numerator and$a+2b$ as its denominator. The second fraction has$7a+14b$ as its numerator and$m+3$ as its denominator. -
Multiply the Numerators: Multiply
$(m^2+8m+15)$ and$(7a+14b)$ together. This gives us$(7a+14b)(m^2+8m+15)$ as the numerator of the product. -
Multiply the Denominators: Multiply
$(a+2b)$ and$(m+3)$ together. This gives us$(a+2b)(m+3)$ as the denominator of the product. -
Write the Product as a Single Fraction: The product of the two fractions is then
$\frac{(7a+14b)(m^2+8m+15)}{(a+2b)(m+3)}$ .
This is the simplified form of the product of the two given rational expressions. It's important to note that we cannot simplify this expression further without knowing the values of
Here are the steps to multiply the rational expressions
-
Identify the Numerator and Denominator of Each Fraction: The first fraction has
$6r+3$ as its numerator and$r+6$ as its denominator. The second fraction has$r^2+9r+18$ as its numerator and$2r+1$ as its denominator. -
Multiply the Numerators: Multiply
$(6r+3)$ and$(r^2+9r+18)$ together. This gives us$(6r+3)(r^2+9r+18)$ as the numerator of the product. -
Multiply the Denominators: Multiply
$(r+6)$ and$(2r+1)$ together. This gives us$(r+6)(2r+1)$ as the denominator of the product. -
Write the Product as a Single Fraction: The product of the two fractions is then
$\frac{(6r+3)(r^2+9r+18)}{(r+6)(2r+1)}$ .
This is the simplified form of the product of the two given rational expressions. It's important to note that we cannot simplify this expression further without knowing the value of
Here are the steps to divide the rational expressions
-
Identify the Numerator and Denominator of Each Fraction: The first fraction has
$x$ as its numerator and$x+2$ as its denominator. The second fraction has$x^2$ as its numerator and$x^2+6x+8$ as its denominator. -
Reciprocal of the Second Fraction: When dividing fractions, we multiply by the reciprocal of the second fraction. The reciprocal of
$\frac{x^2}{x^2+6x+8}$ is$\frac{x^2+6x+8}{x^2}$ . -
Multiply the Fractions: Multiply
$\frac{x}{x+2}$ and$\frac{x^2+6x+8}{x^2}$ . This gives us$\frac{x(x^2+6x+8)}{x^2(x+2)}$ as the result. -
Simplify the Result: The
$x^2$ in the numerator and denominator cancel out, leaving us with$\frac{x^2+6x+8}{x(x+2)}$ .
This is the simplified form of the quotient of the two given rational expressions. It's important to note that we cannot simplify this expression further without knowing the value of
Here are the steps to add and subtract the rational expressions
-
Identify Like Terms: In this case,
$\frac{2}{x}$ and$\frac{3}{2x}$ are like terms because they have the same variable part,$x$ in the denominator. The term$-\frac{1}{4}$ is a constant and does not have a variable part. -
Add/Subtract Like Terms: Add the coefficients of the like terms together. The coefficient of
$\frac{2}{x}$ is$2$ and the coefficient of$\frac{3}{2x}$ is$\frac{3}{2}$ . Adding these together gives$2 + \frac{3}{2} = \frac{7}{2}$ , so the sum of these two terms is$\frac{7}{2x}$ . -
Combine the Results: Combine the result from step 2 with the constant term
$-\frac{1}{4}$ . This gives the final result of$-\frac{1}{4} + \frac{7}{2x}$ .
This is the simplified form of the sum of the three given rational expressions. It's important to note that we cannot simplify this expression further without knowing the value of
The difference of the rational expressions
This is the simplified form of the difference of the two given rational expressions. Now, let's simplify further:
-
Identify Common Denominator: The common denominator of the two fractions is
$(x-2)(x+2)$ . -
Rewrite the Second Fraction: Rewrite the second fraction
$\frac{3}{x-2}$ as$\frac{3(x+2)}{(x-2)(x+2)}$ to have the common denominator. -
Subtract the Fractions: Now that the fractions have the same denominator, subtract the numerators:
$\frac{4x+1-3(x+2)}{(x-2)(x+2)}$ . -
Simplify the Numerator: Simplify the numerator to get the final result.
Without knowing the value of
The difference of the rational expressions
This is the simplified form of the difference of the two given rational expressions. Now, let's simplify further:
-
Identify Common Denominator: The common denominator of the two fractions is
$x^2+3x-10$ . -
Rewrite the Second Fraction: Rewrite the second fraction
$\frac{2}{x+5}$ as$\frac{2(x-2)}{x^2+3x-10}$ to have the common denominator. -
Subtract the Fractions: Now that the fractions have the same denominator, subtract the numerators:
$-\frac{2(x-2)}{x^2+3x-10} + \frac{3-4x}{x^2+3x-10}$ . -
Simplify the Numerator: Simplify the numerator to get the final result.
Without knowing the value of
The sum of the rational expressions
This is the simplified form of the sum of the two given rational expressions. Now, let's simplify further:
-
Identify Common Denominator: The common denominator of the two fractions is
$x-3$ . -
Rewrite the Second Fraction: Rewrite the second fraction
$\frac{4}{2x-6}$ as$\frac{2 \cdot 4}{2(x-3)}$ to have the common denominator. -
Add the Fractions: Now that the fractions have the same denominator, add the numerators:
$-\frac{3x}{x-3} + \frac{2 \cdot 4}{2(x-3)}$ . -
Simplify the Numerator: Simplify the numerator to get the final result.
Without knowing the value of
The sum of the rational expressions
This is the simplified form of the sum of the two given rational expressions. Now, let's simplify further:
-
Identify Common Denominator: The common denominator of the two fractions is
$x^2-x-6$ and$x^2+4x+4$ . -
Rewrite the Fractions: Rewrite the fractions as
$\frac{5}{-6 - x + x^2}$ and$\frac{4}{4 + 4x + x^2}$ to have the common denominator. -
Add the Fractions: Now that the fractions have the same denominator, add the numerators.
-
Simplify the Numerator: Simplify the numerator to get the final result.
Without knowing the value of
Throughout the module, there will be quizzes to test your knowledge on the topics we have covered. These quizzes will help you assess your understanding and identify areas where you may need to review.
During the virtual lecture, we will use active learning strategies to engage with the material and deepen our understanding. These strategies may include group discussions, problem-solving activities, and interactive simulations.
By the end of this module, you should have a strong foundation in Algebra 1 and be able to apply your knowledge to solve real-world problems.