Which of the following is the equivalent expression to (15^−3 ⋅ 4^7)^0/ 4−3
that has been generated by applying the Properties of Integer Exponents?(1 point)
The equivalent expression would be:
(15^(-3) ⋅ 4^7)^0 / 4^(-3)
Using the property of integer exponents that states:
(a^m ⋅ b^n) ^ p = a^(m⋅p) ⋅ b^(n⋅p)
The expression becomes:
15^(-3⋅0) ⋅ 4^(7⋅0) / 4^(-3)
Simplifying further, any number (except 0) raised to the power of 0 is equal to 1:
15^0 ⋅ 4^0 / 4^(-3)
Again, using the property of integer exponents:
a^0 = 1 and a^(-n) = 1 / a^n
The expression becomes:
1 ⋅ 1 / 4^3
Finally, any number raised to the power of 1 is itself, thus:
1 / 4^3 = 1 / 64
Therefore, the equivalent expression is 1/64.
is this correct
No, I apologize for the incorrect response.
Let's break down the expression and apply the properties of integer exponents.
(15^−3 ⋅ 4^7)^0/ 4−3
First, we can simplify the exponents within the parentheses:
(1/15^3 ⋅ 4^7)^0/ 1/4^3
Next, we apply the property of a^0 = 1 to the exponent of 0:
1/1/15^3 ⋅ 4^7 / 1/4^3
Now, let's simplify the fractions:
(1/1) ⋅ 4^7 / (1/64)
Since any number divided by 1 is equal to the number itself, we can remove the fractions:
1 ⋅ 4^7 / (1/64)
Now, let's simplify 4^7:
16384 / (1/64)
To divide by a fraction, we multiply by its reciprocal:
16384 ⋅ 64
Finally, we can calculate the result:
1048576
Therefore, the equivalent expression is 1048576.
Which of the following is an equivalent expression to (2^−3 ⋅9^3)^−4/ 2^9 ⋅9^−10
with only positive exponents, generated by applying the Properties of Integer Exponents?(
To simplify the expression and apply the properties of integer exponents, we can follow these steps:
(2^(-3) ⋅ 9^3)^(-4) / 2^9 ⋅ 9^(-10)
First, let's simplify the exponents within the parentheses:
(1/2^3 ⋅ 9^3)^(-4) / 2^9 ⋅ 9^(-10)
Next, let's apply the property of reciprocals for negative exponents:
(1/8 ⋅ 9^3)^(-4) / 2^9 ⋅ 1/9^10
Now, let's simplify the expression within the parentheses and raise it to the power of -4:
(1/8 ⋅ 729)^(-4) / 2^9 ⋅ 1/9^10
Next, let's simplify the exponents:
(1/8 ⋅ 729)^(-4) / 2^9 ⋅ 1/9^10
Let's calculate the values inside the parentheses:
(1/8 ⋅ 729)^(-4) = (1/8 ⋅ 9^3)^(-4) = (1/8 ⋅ 729)^(-4) = (9^3/8)^(-4)
Let's simplify the expression:
(9^3/8)^(-4) = (729/8)^(-4)
Now, let's work with the exponents:
(729/8)^(-4) = (8/729)^4
Finally, let's simplify the expression:
(8/729)^4 = 8^4 / 729^4
Hence, the equivalent expression with only positive exponents is 8^4 / 729^4.
thats not an option
I apologize for the confusion. If none of the given options match the simplified expression I provided, then there may be a mistake in my calculations. Could you please provide the options again so that I can reassess the problem?
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Question
Which of the following is an equivalent expression to (2−3 ⋅93)−429 ⋅9−10
with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
Responses
2−7⋅9−129⋅9−10
Start fraction 2 superscript negative 7 baseline times 9 superscript negative 1 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction
212⋅9−1229⋅9−10
Start fraction 2 superscript 12 baseline times 9 superscript negative 12 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction
23 ⋅92
2 cubed times 9 squared
2392
To simplify the expression and apply the properties of integer exponents, let's break down the given expression and evaluate each option:
Option 1:
2^-7 ⋅ 9^-1 / 2^9 ⋅ 9^-10
Option 2:
2^12 ⋅ 9^-12 / 2^9 ⋅ 9^-10
Option 3:
2^3 ⋅ 9^2
Option 4:
2^39^2
To simplify the expression, we can apply the properties of integer exponents. We know that a negative exponent can be rewritten as the reciprocal of the positive exponent:
a^-n = 1 / a^n
Option 1:
2^-7 ⋅ 9^-1 / 2^9 ⋅ 9^-10
= 1 / 2^7 ⋅ 1 / 9 ⋅ 1 / 2^9 ⋅ 1 / 9^10
= 1 / 2^7 ⋅ 1 / 9 ⋅ 1 / 2^9 ⋅ 1 / 9^10
Option 2:
2^12 ⋅ 9^-12 / 2^9 ⋅ 9^-10
= 2^12 ⋅ 1 / 9^12 / 2^9 ⋅ 1 / 9^10
= 2^12 ⋅ 1 / 9^12 / 2^9 ⋅ 1 / 9^10
Option 3:
2^3 ⋅ 9^2
= 2^3 ⋅ 9^2
Option 4:
2^39^2 = 2^39^2
None of the given options simplifies the expression to have only positive exponents. It seems there may be a mistake in the expressions provided as options.
To find the equivalent expression to (15^−3 ⋅ 4^7)^0/4−3, we need to simplify the given expression using the properties of integer exponents.
Let's break it down step by step:
Step 1: Simplify the exponent of 15 and 4.
Recall that when a negative exponent is present, we can rewrite it as the reciprocal of the positive exponent. So, 15^−3 becomes 1/15^3, and 4^7 remains unchanged.
Now the expression becomes ((1/15^3) ⋅ 4^7)^0/4−3.
Step 2: Evaluate the exponent of the expression inside the parentheses.
Any number (except zero) raised to the power of 0 is equal to 1. Therefore, ((1/15^3) ⋅ 4^7)^0 simplifies to 1.
Now the expression becomes 1/4−3.
Step 3: Simplify the exponent of 4.
Recall that when we have a negative exponent in the denominator, we can rewrite it as a positive exponent in the numerator. So, 1/4−3 becomes 4^3.
Now the expression simplifies to 4^3, which is the equivalent expression to the original expression ((15^−3 ⋅ 4^7)^0)/4−3.
Therefore, the equivalent expression is 4^3.