Question

Which expression is equivalent to three to the power of 2×3 to the power of -5

A. 1/3 to he power of 3
B. 1/3 to the power of 7
C. 1/3 to the power of -3
D. 1/3 to the power of -7

Answer 1

The expression equivalent to
`\(3^{2 * 3^(-5)}\)` is
`\(1/3^(-3)\)`, corresponding to option C. (option C)

Step-by-step explanation:

To find the equivalent expression, we need to simplify the given expression step by step. Starting with
`\(3^{2 * 3^(-5)}\)`, we apply the rule
`\(a^(m * n) = a^(mn)\)`, which leads to
`\(3^(2 * -5)\)`. Further simplifying,
`\(2 * -5 = -10\)`, so we have
`\(3^(-10)\).` Now, applying the rule
`\(a^(-n) = 1/a^n\)`, the expression becomes
`\(1/3^(10)\)`. Simplifying further,
`\(1/3^(10) = 1/3^(2 * 5) = 1/3^(-3)\)`.

Therefore, the correct equivalent expression is
`\(1/3^(-3)\)`, which corresponds to option C.

It's crucial to be familiar with the laws of exponents and understand how to manipulate expressions using these rules. In this case, recognizing the power of a power rule
`(\(a^(m^n) = a^(mn)\))` and the negative exponent rule
`(\(a^(-n) = 1/a^n\))` is essential. The correct choice, option C, reflects the simplified form of the given expression.(option C)