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Chapter 1: Problem 15

Evaluate each expression without using a calculator. $$ \left[\left(\frac{2}{3}\right)^{-2}\right]^{-1} $$

Short Answer

Expert verified
The expression evaluates to \( \frac{4}{9} \).

Step by step solution

01

Understand the Inner Exponent

The expression given is \( \left( \frac{2}{3} \right)^{-2} \). When a base is raised to a negative exponent, it implies taking the reciprocal of the base raised to the positive of that exponent. This converts \( \left( \frac{2}{3} \right)^{-2} \) to \( \left( \frac{3}{2} \right)^2 \).
02

Evaluate the Square

Now, evaluate \( \left( \frac{3}{2} \right)^2 \). Squaring a fraction involves squaring both the numerator and the denominator, so \( \left( \frac{3}{2} \right)^2 = \frac{3^2}{2^2} = \frac{9}{4} \).
03

Address the Outer Exponent

The expression \( \left[ \left( \frac{2}{3} \right)^{-2} \right]^{-1} \) simplifies to \( \left( \frac{9}{4} \right)^{-1} \). Raising a number to the power of \(-1\) means taking its reciprocal, so \( \left( \frac{9}{4} \right)^{-1} = \frac{4}{9} \).
04

Final Answer

The evaluated expression \( \left[ \left( \frac{2}{3} \right)^{-2} \right]^{-1} \) simplifies to \( \frac{4}{9} \).

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Key Concepts

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

Reciprocal
A reciprocal of a number is what you get when you flip it upside down! For example, the reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). It's as simple as switching the numerator and the denominator places. Why is this useful? Well, there's a neat trick with exponents: raising a number to the power of \(-1\) is the same as finding its reciprocal.
  • Example: The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
  • Remember: multiplying a number by its reciprocal always gives 1.
Learning about reciprocals helps in understanding how to simplify expressions, especially those with negative exponents.
Negative Exponent
Negative exponents can seem tricky at first, but they are merely another way to express reciprocals. When you see a negative exponent, think reciprocal!
A negative exponent tells us how many times to divide by the number, rather than multiply. For example, \( a^{-n} \) represents \( \frac{1}{a^n} \).
  • Example: \( 2^{-3} \) is the same as \( \frac{1}{2^3} = \frac{1}{8} \).
  • For fractions: \( \left( \frac{b}{a} \right)^{-n} = \left( \frac{a}{b} \right)^n \).
Understanding negative exponents is crucial in simplifying expressions, just like in our exercise.
Fraction
Fractions are numbers that represent parts of a whole. In mathematical terms, a fraction is expressed as \( \frac{numerator}{denominator} \). These play a big role in our calculations鈥攅specially when exponents are involved.
  • The numerator (top number) tells how many parts we have.
  • The denominator (bottom number) tells into how many parts the whole is divided.
Fractions can also be raised to powers, affecting both the numerator and the denominator.
For instance, squaring the fraction \( \frac{3}{2} \) involved squaring both 3 and 2, leading to \( \frac{9}{4} \).
Mastering fractions gives you a strong foundation for handling expressions with exponents.
Power of a Number
Raising a number to a power refers to multiplying it by itself a certain number of times. The power is indicated by the exponent. Exponents tell us how many times the base is used as a factor in multiplication.
  • Example: \( 2^3 \) means \( 2 \times 2 \times 2 \) which equals 8.
  • For fractions, both the numerator and denominator must be raised to the power.
In our exercise, we squared the fraction \( \frac{3}{2} \) as \( \frac{3^2}{2^2} \), which results in \( \frac{9}{4} \).
Understanding powers and exponents helps you simplify and evaluate expressions with ease.

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Most popular questions from this chapter

GENERAL: Impact Velocity If a marble is dropped from a height of \(x\) feet, it will hit the ground with velocity \(v(x)=\frac{60}{11} \sqrt{x}\) miles per hour (neglecting air resistance). Use this formula to find the velocity with which a marble will strike the ground if it is dropped from the height of the tallest building in the United States, the 1776 -foot One World Trade Center in New York City.

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

True or False: If \(f(x)=m x+b,\) then \(f(x+h)=f(x)+m h\).

ALLOMETRY: Heart Rate It is well known that the hearts of smaller animals beat faster than the hearts of larger animals. The actual relationship is approximately (Heart rate) \(=250(\text { Weight })^{-1 / 4}\) where the heart rate is in beats per minute and the weight is in pounds. Use this relationship to estimate the heart rate of: A 625 -pound grizzly bear.

GENERAL: Boiling Point At higher altitudes, water boils at lower temperatures. This is why at high altitudes foods must be boiled for longer times - the lower boiling point imparts less heat to the food. At an altitude of \(h\) thousand feet above sea level, water boils at a temperature of \(B(h)=-1.8 h+212\) degrees Fahrenheit. Find the altitude at which water boils at 98.6 degrees Fahrenheit. (Your answer will show that at a high enough altitude, water boils at normal body temperature. This is why airplane cabins must be pressurized - at high enough altitudes one's blood would boil.)
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