If `(32)/(500) = (2)^(3)/(5)^(m)`, then the value of m is

Options

A. 2
B. 3
C. 4
D. 0

Answer

To solve the equation \(\frac{32}{500} = \frac{2^3}{5^m}\), we need to find the value of \(m\). Let's break it down step by step.

### Step 1: Factor the Numerator and Denominator
First, we will factor the numbers in the fraction on the left side.

- **Numerator**: \(32\) can be expressed as \(2^5\) because \(32 = 2 \times 2 \times 2 \times 2 \times 2\).
- **Denominator**: \(500\) can be factored as follows:
\[ 500 = 5 \times 100 = 5 \times (10 \times 10) = 5 \times (2 \times 5) \times (2 \times 5) = 5^3 \times 2^2 \]

So, we can rewrite the fraction:
\[ \frac{32}{500} = \frac{2^5}{5^3 \times 2^2} \]

### Step 2: Simplify the Fraction
Now we can simplify the fraction:
\[ \frac{2^5}{5^3 \times 2^2} = \frac{2^5}{2^2} \times \frac{1}{5^3} = 2^{5-2} \times 5^{-3} = 2^3 \times 5^{-3} \]

### Step 3: Set the Equation
Now we have:
\[ \frac{32}{500} = 2^3 \times 5^{-3} \]
And we know from the original equation:
\[ \frac{32}{500} = \frac{2^3}{5^m} \]

### Step 4: Compare the Exponents
Since both sides of the equation represent the same fraction, we can equate the powers of \(5\):
\[ 5^{-3} = 5^{-m} \]

### Step 5: Solve for \(m\)
Since the bases are the same, we can set the exponents equal to each other:
\[ -m = -3 \]
Thus, we find:
\[ m = 3 \]

### Final Answer
The value of \(m\) is \(3\).

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