Exponent Rules Unit Test
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Question
Use the Product Rule of Exponents to simplify 510⋅55
To simplify 510⋅55 using the Product Rule of Exponents, we can add the exponents of the same base number, 5.
Following the Product Rule, we get:
510⋅55 = (5^10)⋅(5^5)
= 5^(10+5)
= 5^15
So, 510⋅55 simplifies to 5^15.
What is the missing exponent in the following equation?
h450h? = h215 (1 point)
To find the missing exponent in the equation h450h? = h215, we can use the Quotient Rule of Exponents. According to the Quotient Rule, when dividing two numbers with the same base, we subtract the exponents.
In this case, we have:
h450h? = h215
The base is 'h'. Since the exponents are missing, let's call them 'x' and 'y'.
Using the Quotient Rule, we can write the equation as:
h^(450-x) = h^215
To solve for the missing exponent 'y', we need to set the exponents on both sides of the equation equal to each other.
450 - x = 215
Now, we can solve for 'x':
450 - x = 215
Subtracting 450 from both sides, we get:
-x = -235
Dividing both sides by -1 (or multiplying by -1), we find:
x = 235
Therefore, the missing exponent in the equation h450h? = h215 is 235.
Simplify 1287÷1225 .
To simplify 1287 ÷ 1225, you can simply divide the numerator (1287) by the denominator (1225).
1287 ÷ 1225 = 1.05
Therefore, 1287 ÷ 1225 simplifies to 1.05.
What is the value of 6.9−20 ?(1 point)
Responses
6.9
6.9
5.9
5.9
7.9
7.9
4.9
To calculate the value of 6.9 - 20, you subtract 20 from 6.9:
6.9 - 20 = -13.1
Therefore, the value of 6.9 - 20 is -13.1.
What is the value of 6.9−2^0 ?(1 point)
Responses
6.9
6.9
5.9
5.9
7.9
7.9
4.9
To find the value of 6.9 - 2^0, we need to evaluate the exponent expression first.
2^0 is equal to 1, since any number raised to the power of 0 is equal to 1.
Therefore, the expression simplifies to:
6.9 - 1 = 5.9
Thus, the value of 6.9 - 2^0 is 5.9.
Apply the Zero Power Rule to create an equivalent expression to 9⋅(−33)0+32(8−270).(2 points)
9⋅(−33)0+32(8−270) =
⋅
+32(
–
)
To apply the Zero Power Rule, we need to evaluate any term with an exponent of 0.
In this case, (-33)^0 is equal to 1, since any non-zero number raised to the power of 0 is equal to 1.
Using the Zero Power Rule, the expression 9⋅(-33)^0 + 32(8-270) can be simplified to:
9⋅1 + 32(8-270) =
9 + 32(8-270)
Thus, the equivalent expression using the Zero Power Rule is 9 + 32(8-270).
Apply the Zero Power Rule to create an equivalent expression to 9⋅(−33)0+32(8−270).(2 points)
9⋅(−33)0+32(8−270) =
⋅
+3/2(
–
)
I apologize for any confusion, but it seems there may have been an error in the expression you provided. According to the Zero Power Rule, any non-zero number raised to the power of 0 is equal to 1. However, you wrote (-33)^0 instead of (-3)^0.
Assuming you meant to write 9⋅(-3)^0+32(8−270), we can apply the Zero Power Rule to simplify the expression:
9⋅(-3)^0 + 32(8-270) = 9⋅1 + 32(8-270)
= 9 + 32(8-270)
Thus, the equivalent expression using the Zero Power Rule is 9 + 32(8-270).
Solve the expression 15⋅805+4(70−6) .(1 point)
15⋅805+4(70−6)=
To solve the expression 15⋅805+4(70−6), we need to perform the operations according to the order of operations (also known as PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right).
First, we simplify the expression inside the parentheses:
70 - 6 = 64
Now, we substitute this value back into the original expression:
15⋅805 + 4(64)
Next, we perform the multiplication:
15⋅805 + 4⋅64 = 12075 + 256
Finally, we perform the addition:
12075 + 256 = 12331
Therefore, 15⋅805 + 4(70−6) simplifies to 12331.
Solve the expression 15⋅8^0/5+4(70−6) .(1 point)
15⋅805+4(7^0−6)=
To solve the expression 15⋅8^0/5+4(70−6), we need to perform the operations according to the order of operations (also known as PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right).
First, we simplify the exponent expression:
8^0 is equal to 1, since any number raised to the power of 0 is equal to 1.
Now, we substitute this value into the expression and perform the multiplication and subtraction inside the parentheses:
15⋅1/5 + 4(70-6) = 15/5 + 4(64)
Next, we continue to simplify:
15/5 simplifies to 3, since dividing 15 by 5 equals 3.
Now, we perform the multiplication:
3 + 4(64) = 3 + 256
Finally, we perform the addition:
3 + 256 = 259
Therefore, 15⋅8^0/5+4(70−6) simplifies to 259.
Use the Power Rule of Exponents to simplify (10^3)^6 .
To simplify (10^3)^6 using the Power Rule of Exponents, we need to multiply the exponents of the same base.
According to the Power Rule of Exponents, (a^m)^n is equal to a^(m*n).
In this case, we have (10^3)^6. The base is 10 and the exponents are 3 and 6.
Using the Power Rule, we can simplify this expression as:
(10^3)^6 = 10^(3*6) = 10^18
Therefore, (10^3)^6 simplifies to 10^18.
If the expression 7^15 was simplified using the Power Rule of Exponents, which of the following would be an expanded equivalent expression?(1 point)
Responses
(7⋅7⋅7⋅7⋅7)10
left parenthesis 7 times 7 times 7 times 7 times 7 right parenthesis superscript 10 baseline
(710)5
left parenthesis 7 superscript 10 baseline right parenthesis superscript 5 baseline
(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)
left parenthesis 7 times 7 times 7 right parenthesis times left parenthesis 7 times 7 times 7 right parenthesis times left parenthesis 7 times 7 times 7 right parenthesis times left parenthesis 7 times 7 times 7 right parenthesis times left parenthesis 7 times 7 times 7 right parenthesis
(718)3
If the expression 7^15 is simplified using the Power Rule of Exponents, an expanded equivalent expression would be (7⋅7⋅7⋅7⋅7)10, which reads as "7 times 7 times 7 times 7 times 7, all raised to the power of 10."
Therefore, the correct response is: (7⋅7⋅7⋅7⋅7)10.
Use the Power Rule of Exponents to create a simplified equivalent expression for (19^4)^9
To simplify (19^4)^9 using the Power Rule of Exponents, we need to multiply the exponents of the same base.
According to the Power Rule of Exponents, (a^m)^n is equal to a^(m*n).
In this case, we have (19^4)^9. The base is 19 and the exponents are 4 and 9.
Using the Power Rule, we can simplify this expression as:
(19^4)^9 = 19^(4*9) = 19^36
Therefore, (19^4)^9 simplifies to 19^36.
Apply the Property of Negative Integer Exponents to solve 7−^3 . The answer should not contain any exponent expression or decimals.(1 point)
7−^3=
To apply the Property of Negative Integer Exponents, we know that a negative exponent represents the reciprocal of the corresponding positive exponent.
In this case, 7^(-3) can be rewritten as:
1 / 7^3
Now, we can find the value of 7^3, which is equal to:
7^3 = 7 * 7 * 7 = 343
Therefore, 7^(-3) can be simplified as:
1 / 343
So, the answer to 7^(-3) is 1/343.
Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3−178−14 with positive exponents only.(1 point)
3−178−14=
To apply the Property of Negative Integer Exponents, we can use the rule that states a negative exponent can be rewritten as the reciprocal of the corresponding positive exponent.
In this case, we have 3^(-17) * 8^(-14). We can rewrite this expression by taking the reciprocal of each base and changing the sign of the exponent:
(1/3^17) * (1/8^14)
Now, we can simplify further by using the rule that states a product of reciprocals is equal to 1:
1 / (3^17 * 8^14)
Therefore, the expression 3^(-17) * 8^(-14) with positive exponents only is 1 / (3^17 * 8^14).
Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3−^17/8−^14 with positive exponents only.(1 point)
3−^17/8−^14=
To apply the Property of Negative Integer Exponents, we can use the rule that states a negative exponent can be rewritten as the reciprocal of the corresponding positive exponent.
In this case, we have 3^(-17) / 8^(-14). We can rewrite this expression by taking the reciprocal of each base and changing the sign of the exponent:
(1 / 3^17) / (1 / 8^14)
Now, we can simplify further by using the rule that dividing by a fraction is equivalent to multiplying by its reciprocal:
1 / (3^17) * (8^14 / 1)
Therefore, the expression 3^(-17) / 8^(-14) with positive exponents only is 1 / (3^17) * (8^14).
Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)7 ⋅ 7206−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
(7−3)7 ⋅ 7206−2=
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can use the following rules:
1. A negative exponent can be rewritten as the reciprocal of the corresponding positive exponent.
2. The product of two numbers with the same base raised to different exponents can be simplified by adding the exponents.
Starting with the expression (7^-3)7 ⋅ 720^6^-2, let's apply these rules:
(7^-3)7 ⋅ 720^6^-2 = (1/7^3)7 ⋅ (1/720^6^2)
Now, let's simplify further:
(1/7^3)7 ⋅ (1/720^6^2) = 1/7^3 * 7/720^6^2
To multiply the fractions, we can multiply the numerators and the denominators separately:
1/7^3 * 7/720^6^2 = (1*7)/(7^3*720^6^2)
Simplifying the denominator:
(1*7)/(7^3*720^6^2) = 7/(7^3*720^6^2)
Now, we simplify further by adding the exponents of 7:
7/(7^3*720^6^2) = 7/(7^3*720^12)
Simplifying 7^3:
7/(7^3*720^12) = 7/(343*720^12)
Now we have the expression 7/(343*720^12).
To solve this expression, we need to simplify it further:
7/(343*720^12) = 7/((7^3)*(2^6*3*5)^12)
Using the rule of exponents, we can simplify the denominator:
7/((7^3)*(2^6*3*5)^12) = 7/((7^3)*(2^72*3^12*5^12))
Since the base numbers are different and cannot be simplified further, the expression 7/(343*720^12) is already in simplest form.
Therefore, the answer to (7^-3)7 ⋅ 720^6^-2, with positive exponents only, is 7/(7^3*720^12), which is already in simplest form, as indicated before.
Simplify the expression (4x^3)^2 using the properties of integer exponents.(1 point)
Responses
4x6
4 x superscript 6 baseline
16x5
16 x superscript 5 baseline
8x6
8 x superscript 6 baseline
16x6
16 x superscript 6 baseline
To simplify the expression (4x^3)^2 using the properties of integer exponents, we apply the Power Rule of Exponents which states that (a^b)^c is equal to a^(b*c).
In this case, we have (4x^3)^2. We can apply the Power Rule by multiplying the exponent inside the parentheses with the exponent outside the parentheses:
(4x^3)^2 = 4^(2*1) * (x^3)^2
Simplifying further, we have:
4^2 * x^(3*2) = 16x^6
Therefore, the expression (4x^3)^2 simplifies to 16x^6.
Apply the properties of integer exponents to generate an equivalent algebraic expression to (4x4y−4)3(x7y3)2 .(1 point)
Responses
64x20
Start Fraction 64 over x superscript 20 baseline End Fraction
x2y1864
Start Fraction x squared y superscript 18 baseline over 64 End Fraction
4x2y18
Start Fraction 4 over x squared y superscript 18 baseline End Fraction
64x2y18
To apply the properties of integer exponents and generate an equivalent algebraic expression to (4x^4y^-4)^3(x^7y^3)^2, we can use the following properties:
1. (a^m)^n = a^(m*n)
2. When multiplying powers with the same base, you add the exponents.
Applying these properties, we can simplify the expression:
(4x^4y^-4)^3(x^7y^3)^2 = 4^3(x^4)^3(y^-4)^3(x^7)^2(y^3)^2
Simplifying each term within the parentheses:
4^3(x^4)^3(y^-4)^3(x^7)^2(y^3)^2 = 64x^12y^-12x^14y^6
Now, combining the terms with the same base:
64x^12y^-12x^14y^6 = 64x^(12+14)y^(-12+6)
Simplifying the exponents:
64x^(12+14)y^(-12+6) = 64x^26y^-6
Finally, rearranging the expression to have positive exponents:
64x^26y^-6 = 64(x^26 / y^6)
Therefore, the equivalent algebraic expression to (4x^4y^-4)^3(x^7y^3)^2 is 64(x^26 / y^6