/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 \(\frac{7}{7^{1 / 3}}\)... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 6

\(\frac{7}{7^{1 / 3}}\)

Short Answer

Expert verified
1

Step by step solution

01

Write Out Expression

Rewrite the expression into \(\frac{7}{7^{1 / 3}}\), which means the cube root of 7.
02

Simplify Fraction

We know that \(\frac{a}{b}\) is the same as a multiplied by the reciprocal of b, 1/b. So, we will write \(\frac{7}{7^{1 / 3}}\) as \(7*7^{-1 / 3}\).
03

Apply the Power of a Power Rule

Using the power of a power rule we can simplify the expression as \(7*7^{(-1 / 3)*3}\). Applying the rule, we have \(7*7^{-1}\).
04

Simplify Further

Now we apply the negative exponent rule to get \(7* (1/7)\).
05

Final Simplification

Lastly, we multiply the fractions to get final answer 1.

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Key Concepts

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

Negative Exponents
Negative exponents can be a tricky concept at first, but they become straightforward once you understand their purpose. The negative exponent tells you to take the reciprocal of the base. For example, in the expression $a^{-n}$, instead of multiplying $a$ by itself $n$ times, you take the reciprocal of $a$, which becomes $1/a^n$. This essentially "flips" the base's position in a fraction.
  • A negative exponent flips the base to the denominator.
  • It tells you to take the reciprocal, or "invert" the base.
  • Negative exponents never make the base negative, only change its position.
In practical terms, if you have $7^{-1}$, this is the same as writing $1/7$.
When simplifying expressions like $ rac{7}{7^{1/3}}$, rewiring them using negative exponents helps.
Rewriting $ rac{7}{7^{1/3}}$ as $7 imes 7^{-1/3}$ allows easier manipulation and simplification.
Rational Exponents
Rational exponents are an alternative way to express roots. They serve the same purpose as a radical. For example, \(a^{m/n}\) represents the \(n\)th root of \(a\) raised to the \(m\)th power, written as \((\sqrt[n]{a})^m\). This notation is particularly helpful in making calculations easier or when simplifying expressions.
  • \(a^{1/n}\) means the \(n\)-th root of \(a\).
  • \(a^{m/n}\) means the \(n\)-th root of \(a\) raised to the \(m\)-power.
  • It simplifies complex root calculations.
In our example, \(7^{1/3}\) can be viewed as the cube root of 7. When rewritten as a fraction with a negative exponent \(7 \times 7^{-1/3}\), it becomes easier to apply further rules of exponents.
Power of a Power Rule
The power of a power rule is a fundamental exponent law that makes it possible to simplify expressions where an exponent is raised to another exponent. According to this rule, \((a^m)^n = a^{m \times n}\). This law helps in collapsing multiple layers of exponents into a single exponent, making expressions more manageable.
  • Multiplying exponents: \((a^m)^n = a^{m \times n}\).
  • Useful for simplifying expressions with nested exponents.
  • Essential for reducing expressions in fewer steps.
In the exercise, \(7 \times 7^{-1/3}\) is simplified using the power of a power rule: \(7 \times 7^{(-1/3)\times3}\).
This step results in the simplification \(7 \times 7^{-1}\).
Using this rule allows for an orderly simplification, which makes the final computation straightforward, as the step becomes simply multiplying by the reciprocal, yielding the answer 1.

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