/*! 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 89 Simplify each expression with ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 89

Simplify each expression with exponents. (a) \(10^{4}\) (b) \(17^{1}\) (c) \(\left(\frac{2}{9}\right)^{2}\) (d )\((0.5)^{3}\)

Short Answer

Expert verified
(a) 10,000; (b) 17; (c) \(\frac{4}{81}\); (d) 0.125

Step by step solution

01

Simplify the first expression

The expression is \(10^{4}\). This means multiply 10 by itself 4 times.\[10 \times 10 \times 10 \times 10 = 10,000\]
02

Simplify the second expression

The expression is \(17^{1}\). Any number to the power of 1 is the number itself.\[17^{1} = 17\]
03

Simplify the third expression

The expression is \(\left(\frac{2}{9}\right)^{2}\). This means multiply \(\frac{2}{9}\) by itself.\[\left(\frac{2}{9}\right)\times\left(\frac{2}{9}\right) = \frac{2\times 2}{9\times 9} = \frac{4}{81}\]
04

Simplify the fourth expression

The expression is \((0.5)^{3}\). This means multiply 0.5 by itself 3 times.\[0.5 \times 0.5 \times 0.5 = 0.125\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

exponentiation
Exponentiation is a mathematical operation, where a number (the base) is multiplied by itself a certain number of times (the exponent). It is denoted as \(a^n\), where \(a\) is the base and \(n\) is the exponent. For example, in the expression \(10^4\), 10 is the base and 4 is the exponent. This means we multiply 10 by itself 4 times: \[10 \times 10 \times 10 \times 10 = 10,000\]The process follows these basic rules:
  • Any number raised to the power of 1 is the number itself: \(a^1 = a\).
  • Any number (except zero) raised to the power of 0 is 1: \(a^0 = 1\).
  • Multiplying a number several times means you keep multiplying the last result by the base number.
Understanding these rules is crucial for simplifying expressions involving exponents.
powers of numbers
The power of a number refers to how many times to use the number in a multiplication. For instance, \(2^3\) means you multiply 2 by itself three times: \[2 \times 2 \times 2 = 8\]Here are some key points to consider:
  • \(a^n\) is read as 'a to the power of n'.
  • If the exponent is positive, like \(5^2\), the multiplication is straightforward as \[5 \times 5 = 25\]
  • If the exponent is negative, say \(5^{-2}\), it means you take the reciprocal of the base raised to the positive exponent: \[5^{-2} = \frac{1}{5^2} = \frac{1}{25}\]
The exercise also involves powers of fractions such as \(\frac{2}{9}^2\). In this case, multiply both the numerator and the denominator by themselves: \[\frac{2}{9} \times \frac{2}{9} = \frac{4}{81}\]Similarly, when dealing with decimals like \(0.5^3\), multiply 0.5 by itself three times: \[0.5 \times 0.5 \times 0.5 = 0.125\]
fractional exponents
Fractional exponents might look tricky, but they represent roots and powers at the same time. For example, \(a^{\frac{1}{2}}\) implies the square root of \(a\). Generally speaking, \(a^{\frac{m}{n}}\) means you first take the n-th root of \(a\) and then raise it to the m-th power, or vice versa.Key insights into fractional exponents:
  • \(a^{\frac{1}{2}} = \text{鈭殅a\)
  • \(a^{\frac{1}{3}} = \text{鈭泒a\)
  • More generally, \(a^{\frac{m}{n}} = (\text{n-th root of } a)^m = (\text{n-th root of } a^m)\)
For example, \(27^{\frac{2}{3}}\) involves first taking the cube root of 27, which is 3, and then squaring the result, so you get \[27^{\frac{2}{3}} = (3)^2 = 9\]
decimal exponents
Decimal exponents are simply another way to write fractional exponents. They can also represent roots and powers, but expressed in decimal form. For example, \(8^{0.333}\) is essentially \(8^{\frac{1}{3}}\), which equals the cube root of 8, giving us 2.Some key points about decimal exponents:
  • Decimal exponents work exactly like fractional exponents.
  • They provide a more precise way to express roots especially when fractional exponents result in repeating decimals.
  • While simplifying expressions with decimal exponents, it's helpful to revert to fractional forms to understand and simplify.
In the given exercise, expressing \(0.5^3\) as a decimal exponent doesn't complicate the calculation: \[0.5^3 \text{ is still } 0.125\]. Calculations involve multiplying 0.5 by itself three times, as previously mentioned.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Divide the monomials. $$\frac{65 a^{10} b^{8} c^{5}}{42 a^{7} b^{6} c^{8}}$$

Divide the monomials. $$\frac{\left(4 u^{2} v^{5}\right)\left(15 u^{3} v\right)}{\left(12 u^{3} v\right)\left(u^{4} v\right)}$$

Divide each polynomial by the binomial. $$\left(q^{2}+2 q+20\right) \div(q+6)$$

Divide each polynomial by the monomial. $$\frac{20 b^{2}-12 b}{-4}$$

Divide each polynomial by the binomial. $$\left(y^{2}+7 y+12\right) \div(y+3)$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.