Question

Simplify \(\frac{3^7}{3^5}\) A. 9 B. \(\frac{1}{9}\) C. 6 D. \(\frac{1}{6}\)

Short answer

A: 9

Step-by-step solution

Understand the Exponents

Recognize that the problem deals with a fraction where both the numerator and the denominator are powers of the same base (3). This means the properties of exponents can be applied.

Apply the Quotient Rule for Exponents

Use the quotient rule for exponents, which states \(\frac{a^m}{a^n} = a^{m-n}\), where a is the base, and m and n are the exponents.

Subtract the Exponents

Subtract the exponent in the denominator from the exponent in the numerator: \(3^{7-5}\)

Simplify the Exponent

Simplify the exponent: \(7-5=2\). This gives us \(3^2\).

Evaluate the Expression

Calculate \(3^2\), which is \(3 \times 3 = 9\).

Choose the Correct Answer

Compare the result to the given options. The correct answer is A: 9.

Key concepts

quotient_rule

When simplifying expressions involving exponents, one of the key rules to know is the quotient rule. This rule is essential when you encounter a fraction where the numerator and denominator are powers of the same base. The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \), where \( a \) is the base and \( m \) and \( n \) are the exponents. In simple terms, this rule tells us that we can subtract the exponent in the denominator from the exponent in the numerator, and the base remains the same.
For example, consider \( \frac{3^7}{3^5} \). Since the base (3) is the same in both the numerator and the denominator, we apply the quotient rule to get \( 3^{7-5} \).

exponent_subtraction

Exponent subtraction is a crucial step when applying the quotient rule. This step simplifies the expression by reducing the exponents. Let's see how it works with our example \( \frac{3^7}{3^5} \).
After applying the quotient rule, we have \( 3^{7-5} \).
Here, we subtract the exponent 5 in the denominator from the exponent 7 in the numerator. This gives us \( 3^2 \), because \( 7-5=2 \).
By subtracting the exponents, we reduce the complexity of the expression, making it easier to evaluate.

evaluating_powers

After simplifying the exponent, the next step is evaluating the power. This means calculating the numerical value of the expression with the simplified exponent. For \( 3^2 \), you need to calculate what 3 raised to the power of 2 equals.
To do this, you multiply the base (3) by itself as many times as the exponent indicates. Since the exponent is 2, we perform the calculation: \( 3 \times 3 = 9 \).
Therefore, evaluating \( 3^2 \) gives us 9.
By following each of these steps - applying the quotient rule, subtracting the exponents, and evaluating the power - we can confidently simplify exponent expressions. This approach not only simplifies your homework but also strengthens your understanding of the fundamental concepts of exponents.