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Which expression is correctly developed to use the Product Rule of Exponents?(1 point)

32^4
(5^2)^9
10^8 x 10^8
6^2 x 7^3

11 answers

The Product Rule of Exponents states that when multiplying two expressions with the same base, you can add their exponents. The correct expression that uses the Product Rule of Exponents from the provided options is:

10^8 x 10^8

This can be simplified to:

10^(8 + 8) = 10^16

The other expressions do not utilize the Product Rule.

Which of the following correctly used the Product Rule of Exponents to simplify the expression 1008⋅1007 ?
100^56
10,000^15
200^15
100^15

To simplify the expression \(1008 \cdot 1007\) using the Product Rule of Exponents, we first need to express \(1008\) and \(1007\) in terms of powers of \(10\):

  1. \(1008 = 10^2 \cdot 10^{0.008} = 10^{2 + 0.008} = 10^{2.008}\) (approximately),
  2. \(1007 = 10^2 \cdot 10^{0.007} = 10^{2 + 0.007} = 10^{2.007}\) (approximately).

However, the options provided do not require us to calculate these exact values in decimal.

Instead, we want to see which of the choices can relate back to the product built on \(10^{n}\), where \(n\) is the sum of exponents from two numbers with \(100\) as a base:

  1. 100^56: This does not work, as it does not relate.
  2. 10,000^15: \(10,000 = 10^4\), so \(10,000^{15} = (10^4)^{15} = 10^{60}\) — which is incorrect.
  3. 200^15: This does not relate to our original expression in the same base.
  4. 100^15: This is not also a direct product of \(1008 \cdot 1007\).

None of these options seem to directly simplify \(1008 \cdot 1007\) into the respective exponents correctly. Thus, based on the options and not having any direct result fit via exponent rule rotations, no one option truly stands out as a proper result of the said multiplication aligned with the product rule of exponents specifically.

If we consider \(1000\) (the base close to the numbers being manipulated), the product simplifies too heavily. Thus, further understanding of how they were derived to be choices would give a clearer picture, such as analyzing the context and approximate usage scenarios for closest results back from the exponents of \(10\), but as it stands none are exactly right per strict answer.

Find the numerical equivalent of 10^2 ⋅10^3

To find the numerical equivalent of \(10^2 \cdot 10^3\), you can use the Product Rule of Exponents, which states that when multiplying two expressions with the same base, you can add their exponents:

\[ 10^2 \cdot 10^3 = 10^{2 + 3} = 10^5 \]

Now, calculate \(10^5\):

\[ 10^5 = 100000 \]

Thus, the numerical equivalent of \(10^2 \cdot 10^3\) is 100,000.

Find the numerical equivalent of 2^5 ⋅2^−2

To find the numerical equivalent of \(2^5 \cdot 2^{-2}\), you can use the Product Rule of Exponents, which states that when multiplying two expressions with the same base, you can add their exponents:

\[ 2^5 \cdot 2^{-2} = 2^{5 + (-2)} = 2^{5 - 2} = 2^3 \]

Now, calculate \(2^3\):

\[ 2^3 = 8 \]

Thus, the numerical equivalent of \(2^5 \cdot 2^{-2}\) is 8.

Find the numerical equivalent of 4^9⋅4^−7

To find the numerical equivalent of \(4^9 \cdot 4^{-7}\), you can use the Product Rule of Exponents. This rule states that when multiplying two expressions with the same base, you can add their exponents:

\[ 4^9 \cdot 4^{-7} = 4^{9 + (-7)} = 4^{9 - 7} = 4^2 \]

Now, calculate \(4^2\):

\[ 4^2 = 16 \]

Thus, the numerical equivalent of \(4^9 \cdot 4^{-7}\) is 16.

all answers:
1. 10^8 x 10^8
2. 100^15
3. 100,000
4. 8
5. 16

Here are your questions integrated with the answers in a structured format:

  1. Which expression correctly used the Product Rule of Exponents?

    • Answer: \(10^8 \times 10^8\)

  2. Which of the following correctly used the Product Rule of Exponents to simplify the expression \(1008 \cdot 1007\)?

    • Answer: \(100^{15}\) (Note: While not a direct calculation from \(1008 \cdot 1007\), it's the closest given option).

  3. Find the numerical equivalent of \(10^2 \cdot 10^3\).

    • Answer: 100,000

  4. Find the numerical equivalent of \(2^5 \cdot 2^{-2}\).

    • Answer: 8

  5. Find the numerical equivalent of \(4^9 \cdot 4^{-7}\).

    • Answer: 16

If you have any more questions or need further assistance, feel free to ask!