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Chapter 1: Problem 43

Evaluate each expression without using a calculator. $$ \left(\frac{25}{16}\right)^{-1 / 2} $$

Short Answer

Expert verified
The value of the expression is \( \frac{4}{5} \).

Step by step solution

01

Understand the Negative Exponent

The expression \( \left( \frac{25}{16} \right)^{-1 / 2} \) involves a negative exponent. A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. This means we need to take the reciprocal of \( \frac{25}{16} \) and raise it to the \( \frac{1}{2} \) power.
02

Calculate the Reciprocal

Taking the reciprocal of \( \frac{25}{16} \) gives us \( \frac{16}{25} \). Now the expression becomes \( \left( \frac{16}{25} \right)^{1/2} \).
03

Evaluate the Square Root

The \( \frac{1}{2} \) exponent indicates taking the square root of the base. So, we need to find \( \sqrt{\frac{16}{25}} \).
04

Simplify the Square Root

The square root of a fraction \( \frac{a}{b} \) is \( \frac{\sqrt{a}}{\sqrt{b}} \). Calculate \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \). Therefore, \( \sqrt{\frac{16}{25}} = \frac{4}{5} \).

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Key Concepts

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

Negative Exponents
Negative exponents might seem tricky at first, but they're not too bad once you get the hang of it. A negative exponent indicates that you need to take the reciprocal of the base. But what is a reciprocal? Let's explore.
  • If you see a negative exponent, like \[x^{-1}\], switch it to its reciprocal form \[\frac{1}{x}\].
  • This means you are essentially "flipping" the base fraction over.
  • For example, in the expression \[\left( \frac{25}{16} \right)^{-1/2}\], start by taking the reciprocal so that it becomes \[\left( \frac{16}{25} \right)^{1/2}\].
The negative exponent step is crucial as it sets the path for simplifying the expression correctly without a calculator.
Reciprocal
The reciprocal is essentially a mathematical term for "opposite" in terms of position. When you flip a fraction, you get its reciprocal. Let's break it down.
  • The reciprocal of a number \(a\) is simply \(\frac{1}{a}\).
  • For fractions, you invert the numerator and the denominator. For instance, the reciprocal of \(\frac{25}{16}\) is \(\frac{16}{25}\).
  • This concept is integral when dealing with negative exponents because it transforms the expression to make it easier to evaluate, such as moving to square roots.
Understanding reciprocals is essential especially when working with expressions involving exponents like in our original exercise. Once you've mastered this, tackling more extensive expressions becomes easier.
Square Root
Square roots might bring back classroom memories where you learned about the magic of finding numbers that multiply by themselves to form another number. In mathematics, they are represented by the \(\sqrt{}\) symbol or a \(\frac{1}{2}\) exponent. Here's how it works!
  • The square root of a number \(x\) is a number \(y\) such that \(y^2 = x\).
  • When given a fractional base, like \(\left(\frac{16}{25}\right)^{1/2}\), you find \(\sqrt{16}\) and \(\sqrt{25}\) separately.
  • These become \(4\) and \(5\), respectively, giving you \(\frac{4}{5}\).
Taking the square root simplifies the expression greatly, paving the way to your final simplified answer. It's as if you are rolling back the effect of squaring a number.
Fraction Simplification
Simplifying fractions is all about reducing them to their most basic form, ensuring they are easy to understand and work with. Let's simplify this process:
  • Start by identifying if the numerator and the denominator have any common factors.
  • If they do, divide both by the greatest common divisor (GCD) to get the simplest form.
  • Applying this to the fraction \(\frac{4}{5}\), you will find it is already in its simplest form, as \(4\) and \(5\) share no common factors other than \(1\).
This step makes fractions more manageable and makes the operation results far neater. It comes in handy in algebra and everyday calculations, ensuring clarity and simplicity in mathematical communication.

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Most popular questions from this chapter

89-90. Some organisms exhibit a density-dependent mortality from one generation to the next. Let \(R>1\) be the net reproductive rate (that is, the number of surviving offspring per parent), let \(x>0\) be the density of parents and \(y\) be the density of surviving offspring. The Beverton-Holt recruitment curve is $$ y=\frac{R x}{1+\left(\frac{R-1}{K}\right) x} $$ where \(K>0\) is the carrying capacity of the environment. Notice that if \(x=K\), then \(y=K\). Show that if \(x

ATHLETICS: Muscle Contraction The fundamental equation of muscle contraction is of the form \((w+a)(v+b)=c\), where \(w\) is the weight placed on the muscle, \(v\) is the velocity of contraction of the muscle, and \(a, b\), and \(c\) are constants that depend upon the muscle and the units of measurement. Solve this equation for \(v\) as a function of \(w, a, b\), and \(c\).

BUSINESS: Break-Even Points and Maximum Profit A company that produces tracking devices for computer disk drives finds that if it produces \(x\) devices per week, its costs will be \(C(x)=180 x+16,000\) and its revenue will be \(R(x)=-2 x^{2}+660 x\) (both in dollars). a. Find the company's break-even points. b. Find the number of devices that will maximize profit, and the maximum profit.

$$ \text { If } f(x)=x+a \text { , then } f(f(x))=\text { ? } $$

BUSINESS: Isocost Lines An isocost line (iso means "same") shows the different combinations of labor and capital (the value of factory buildings, machinery, and so on) a company may buy for the same total cost. An isocost line has equation $$ w L+r K=\mathrm{C} \quad \text { for } L \geq 0, \quad K \geq 0 $$ where \(L\) is the units of labor costing \(w\) dollars per unit, \(K\) is the units of capital purchased at \(r\) dollars per unit, and \(C\) is the total cost. Since both \(L\) and \(K\) must be nonnegative, an isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=8, \quad r=6, \quad C=15,000\), and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. \((1875,0),(1200,900),(600,1700),(0,2500)\)
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