/*! 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 124 \(\sqrt[6]{(-4)^{6}}\)... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 124

\(\sqrt[6]{(-4)^{6}}\)

Short Answer

Expert verified
The simplified result is 4.

Step by step solution

01

- Understand the Problem

We are given the expression \(\sqrt[6]{(-4)^{6}}\) and need to simplify it.
02

- Apply the Property of Radicals and Exponents

Recall the property that \(\root[n]{a^n} = |a|\) when \( n\) is even. In this case, we can use this property because our exponent is 6 (an even number).
03

- Simplify Inside the Radicals

Since \( (-4)^6\) inside the radical stands for the 6th power of -4, we recognize that raising a negative number to an even power results in a positive number. Therefore, \( (-4)^6 = 4^6\).
04

- Apply the Radical

Using the property from Step 2, we find \鈭歔6]{4^6} = |4| = 4. Thus, the simplified result of \(\root[6]{(-4)^6}\) is 4.

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Key Concepts

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

Properties of Exponents
Exponents are a way to express repeated multiplication. For example, \(a^n\) means multiplying \(a\) by itself \(n\) times.
There are several important properties of exponents that you should know:
  • Product of Powers: When multiplying two expressions with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
  • Quotient of Powers: When dividing two expressions with the same base, subtract the exponents: \(a^m \div a^n = a^{m-n}\).
  • Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).
  • Negative Exponent: A negative exponent means taking the reciprocal of the base and making the exponent positive: \(a^{-n} = \frac{1}{a^n} \).
  • Zero Exponent: Any number raised to the zero power is one: \(a^0 = 1 \).
Understanding these properties will help you simplify expressions with exponents, such as the one in our exercise. Remember, for small exponents, it's usually faster to compute directly, but knowing these rules will make complex problems easier.
Properties of Radicals
Radicals are related to exponents but deal with roots instead of powers. The radical symbol \(\root{n}{a}\) represents the \(\root{n}{a}\)-th root of \(a\).
Here are some key properties of radicals:
  • Product Property: \(\root{n}{a \times b} = \root{n}{a} \times \root{n}{b} \).
  • Quotient Property: \(\root{n}{\frac{a}{b}} = \frac{\root{n}{a}}{\root{n}{b}} \)
  • Power of a Radical: \(\root{n}{a^m} = (a^{m/n}) \)
In our exercise, we used the property \(\root{n}{a^n} = |a| \) when \(n\) is even. This is essential because it simplifies radicals with even roots by converting them to absolute values, ensuring the result is non-negative.
Absolute Value
Absolute value measures a number鈥檚 distance from zero on the number line, regardless of direction. It鈥檚 represented by two vertical bars: \(|a|\).
Key points about absolute value:
  • Non-negative: The absolute value is always positive or zero. So, \(|3| = 3\) and \(|-3| = 3\).
  • Distance Interpretation: Think of absolute value as the distance from zero: \( |-4| = 4\) means -4 is 4 units away from zero on the number line.
  • Properties:
    • \(|a \times b| = |a| \times |b|\)
    • \(|a/b| = |a|/|b|\)
    • \(|a+b| \leq |a|+ |b|\). This is called the Triangle Inequality.
In our exercise, we took \(\root{6}{(-4)^6} \) and used the property \( \root{n}{a^n} = |a|\) to simplify it to the absolute value of 4, which is 4. Absolute value ensures that the result of an even root is always non-negative.

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