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Chapter 0: Problem 15

evaluate the expression for the given value of x. $$ x^{-2 / 5} \quad x=-32 $$

Short Answer

Expert verified
The evaluated expression for x=-32 is \(\frac{1}{4}\)

Step by step solution

01

Rewrite the expression in a more comprehensible form

Begin by noting the negative exponent in the expression. Convert \(x^{-2 / 5}\) into \(\frac{1}{x^{2/5}}\) by applying the rule that any base with a negative exponent is equal to 1 divided by the base with the same positive exponent.
02

Substitute the given value of x

Substitute -32 as the value for x into the expression which now gives you \(\frac{1}{(-32)^{2/5}}\).
03

Evaluate the expression

Begin evaluating \((-32)^{2/5}\) by simplifying it as the square root of -32 raised to the power of 2 first and then by taking the fifth root. The square of -32 is \(1024\). The fifth root of \(1024\) is \(4\). Therefore, \((-32)^{2/5}\) equals \(4\). The expression turns out to be \(\frac{1}{4}\).

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Key Concepts

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

Exponent Rules
When dealing with exponents, understanding the basic rules can be really helpful to simplify and evaluate expressions. Exponent rules tell us how to manipulate expressions with powers or indices. Here are a few key rules to keep in mind:
  • The **Product of Powers Rule** states that when multiplying two powers with the same base, we can add the exponents: \(a^m \times a^n = a^{m+n}\).
  • The **Power of a Power Rule** tells us to multiply the exponents when taking a power of a power: \((a^m)^n = a^{m \times n}\).
  • Finally, the **Negative Exponent Rule**, which was used in our solution, states that a base with a negative exponent can be expressed as the reciprocal of that base with a positive exponent: \(a^{-m} = \frac{1}{a^m}\).
By applying these rules, we transformed \(x^{-2/5}\) into \(\frac{1}{x^{2/5}}\). These rules simplify the manipulation of any expression involving exponents.
Fractional Exponents
Fractional exponents provide a nifty way of expressing powers and roots simultaneously. When we see an exponent as a fraction, it indicates both a root and a power. To interpret fractional exponents effectively:
  • The rule \(a^{m/n}\) implies we should raise \(a\) to the power of \(m\) and then take the \(n\)-th root, or vice versa.
  • For instance, in our expression, \((-32)^{2/5}\), we apply the fractional exponent by raising \(-32\) first to the power of \(2\) resulting in 1024, and subsequently taking the fifth root.
What makes fractional exponents particularly useful is how they allow us to use just one notation to signify both powers and roots, instead of switching back and forth with radical symbols.
Expression Evaluation
Evaluating expressions involves the step-by-step application of math rules to find a specific outcome. In our problem, we started with substituting the value of \(x\), which is \(-32\), directly into the expression transforming it to \(\frac{1}{(-32)^{2/5}}\). Next steps include:
  • Determining \((-32)^{2/5}\). We first calculated \((-32)^2 = 1024\).
  • The next step was finding the fifth root of 1024, which is 4. Hence, \((-32)^{2/5} = 4\).
  • Finally, we took the reciprocal because our original exponent was negative, arriving at the result \(\frac{1}{4}\).
Following such steps systematically ensures we get the correct evaluation of an expression.

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Most popular questions from this chapter

Period of a Pendulum The period of a pendulum is \(T=2 \pi \sqrt{\frac{L}{32}}\) where \(T\) is the period in seconds and \(L\) is the length of the pendulum in feet. Find the period of a pendulum whose length is 4 feet.

simplify each expression by factoring. $$ 2 x^{5 / 2}+x^{-1 / 2} $$

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