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Chapter 0: Problem 39

Use the laws of exponents to compute the numbers. $$\frac{7^{4 / 3}}{7^{1 / 3}}$$

Short Answer

Expert verified
7

Step by step solution

01

Use the Quotient Rule of Exponents

The quotient rule of exponents states that \(\frac{a^m}{a^n} = a^{m-n}\). Apply this rule to the given expression: \(\frac{7^{4/3}}{7^{1/3}} = 7^{(4/3) - (1/3)}\).
02

Subtract the Exponents

Perform the subtraction in the exponent: \(7^{(4/3) - (1/3)} = 7^{3/3}\).
03

Simplify the Exponent

Simplify \(7^{3/3}\) by evaluating the fraction \(3/3\), which is equal to 1. Therefore, \(7^{3/3} = 7^1\).
04

Final Answer

Since \(7^1 = 7\), the final answer is 7.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule of Exponents
To solve the given problem, we first use the quotient rule of exponents. This rule states that for any non-zero number 'a', and any integers 'm' and 'n', the expression \(\frac{a^m}{a^n} = a^{m-n}\). Essentially, you subtract the exponent in the denominator from the exponent in the numerator. For example, in the problem: \( \frac{7^{4/3}}{7^{1/3}} \), you can apply the quotient rule to get: \( 7^{(4/3) - (1/3)} \). This step simplifies the problem and is critical in many algebraic manipulations involving exponents. Always remember: same base, different exponents, you subtract.
Simplifying Exponents
After applying the quotient rule, the next step is simplifying the exponents. In our problem: \( 7^{(4/3) - (1/3)}\), you need to perform the subtraction inside the exponent: \( 4/3 - 1/3 \). Since they have the same denominator, subtracting them is straightforward: \(4/3 - 1/3 = 3/3\). Therefore, the expression becomes: \(7^{3/3}\). Simplifying exponents comes down to proper arithmetic on the exponents themselves, and understanding fraction operations is essential here.
Fractional Exponents
Fractional exponents, also known as rational exponents, represent roots. For example, \(a^{1/2}\) is the same as \(\root{2}{a}\), which is the square root of 'a'. In our problem: \(7^{3/3}\), the fraction simplifies to \(1\), but understanding fractional exponents is key. If we had something like \(7^{1/3}\), it means the cubed root of \(7\). So, knowing that \( 7^{3/3} \) simplifies to \(7^1\) because \(3/3\) is 1. Practicing conversions between fractional exponents and roots can be very helpful.

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Most popular questions from this chapter

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