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Chapter 1: Problem 7

Evaluate each expression without using a calculator. $$ \left(\frac{5}{8}\right)^{-1} $$

Short Answer

Expert verified
\( \frac{8}{5} \)

Step by step solution

01

Understand the Negative Exponent

When we have a negative exponent, we take the reciprocal of the base and change the exponent to positive. So, \( \left(\frac{5}{8}\right)^{-1} \) becomes \( \left(\frac{8}{5}\right)^{1} \).
02

Simplify the Expression

Now that we have \( \left(\frac{8}{5}\right)^1 \), we can simplify it. Any number or fraction raised to the power of 1 remains the same, so \( \left(\frac{8}{5}\right)^1 = \frac{8}{5} \).

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Key Concepts

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

Reciprocal
The reciprocal of a number is simply swapping its numerator and denominator. For instance, the reciprocal of \( \frac{5}{8} \) is \( \frac{8}{5} \). Understanding reciprocals is crucial when dealing with negative exponents because a negative exponent indicates the reciprocal of the base number.

Here are key points about reciprocals:
  • If the original number is a fraction, you exchange the places of the numerator and the denominator.
  • If the original number is \( a \), its reciprocal is \( \frac{1}{a} \).
  • The product of a number and its reciprocal is always 1.
Recognizing and applying the concept of reciprocals helps simplify expressions with negative exponents.
Exponentiation
Exponentiation involves raising a number, known as the base, to a certain power. The power, indicated by the exponent, tells us how many times to multiply the base by itself. For example, in \( 2^3 \), 2 is the base and 3 is the exponent, meaning \( 2 \times 2 \times 2 = 8 \).

When dealing with negative exponents, we:
  • Convert the expression to its reciprocal to make the exponent positive.
  • For instance, \( \left(\frac{5}{8}\right)^{-1} \) boils down to calculating \( \left(\frac{8}{5}\right)^{1} \).
  • Understanding the concept of exponentiation is integral to solving complex algebraic expressions.
Being comfortable with this operation builds a strong foundation for more advanced math concepts.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operations. It can be as simple as \( x + 3 \) or more intricate like \( 3x^2 - \frac{5}{x} \). These expressions form the building blocks of algebra.

Within expressions, applying the rules of arithmetic, such as handling exponents, is essential. When you encounter negative exponents inside an algebraic expression, remember:
  • Transform them using reciprocals to turn negative exponents into positive.
  • Simplifying expressions often involves evaluating and reducing using these basic rules.
  • Always double-check to ensure all components of an expression are simplified correctly.
Mastery of algebraic expressions enables you to tackle problems across many areas of mathematics efficiently.

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Most popular questions from this chapter

Which of the following is not a polynomial, and why? $$ x^{2}+\sqrt{2} \quad x^{\sqrt{2}}+1 \quad \sqrt{2} x^{2}+1 $$

If \(f(x)=a x,\) then \(f(f(x))=?\)

ATHLETICS: Juggling If you toss a ball \(h\) feet straight up, it will return to your hand after \(T(h)=0.5 \sqrt{h}\) seconds. This leads to the juggler's dilemma: Juggling more balls means tossing them higher. However, the square root in the above formula means that tossing them twice as high does not gain twice as much time, but only \(\sqrt{2} \approx 1.4\) times as much time. Because of this, there is a limit to the number of balls that a person can juggle, which seems to be about ten. Use this formula to find: a. How long will a ball spend in the air if it is tossed to a height of 4 feet? 8 feet? b. How high must it be tossed to spend 2 seconds in the air? 3 seconds in the air?

An insurance company keeps reserves (money to pay claims) of \(R(v)=2 v^{0.3}\) where \(v\) is the value of all of its policies, and the value of its policies is predicted to be \(v(t)=60+3 t,\) where \(t\) is the number of years from now. (Both \(R\) and \(v\) are in millions of dollars.) Express the reserves \(R\) as a function of \(t,\) and evaluate the function at \(t=10 .\)

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{10 \sqrt{x}}{2 \sqrt[3]{x}} $$
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