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Chapter 7: Problem 31

Simplify each of the following as completely as possible. $$\left(x^{2} y^{3}\right)^{5}$$

Short Answer

Expert verified
Simplified form is \(x^{10} y^{15}\).

Step by step solution

01

Apply the power rule

When raising a power to another power, multiply the exponents. For \(\big(x^{2} y^{3}\big)^{5}\), apply the rule to each factor in the parenthesis separately.
02

Simplify the exponents for each variable

Multiply the exponents for each variable: For \(\big(x^{2}\big)^{5}\) and \(\big(y^{3}\big)^{5}\), calculate the products of the exponents. This gives \(\big(x^{2}\big)^{5} = x^{(2 \times 5)} = x^{10}\) and \(\big(y^{3}\big)^{5} = y^{(3 \times 5)} = y^{15}\).
03

Combine the results

Combine the simplified exponents into a single expression. The expression becomes \(x^{10} y^{15}\).

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Key Concepts

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

Power rule
The power rule is a fundamental concept in algebra that simplifies expressions involving exponents. It states that when you have an exponent raised to another exponent, you multiply the exponents together. For example, in the expression \((x^{a})^{b}\), the power rule tells you to multiply \((a \times b)\), resulting in \(x^{a \times b}\).
Let's apply this rule to the original exercise. In the expression \((x^{2} y^{3})^{5}\), we need to handle both variables separately:
  • For \((x^{2})^{5}\), use the power rule: \(x^{2 \times 5} = x^{10}\).
  • For \((y^{3})^{5}\), use the power rule: \(y^{3 \times 5} = y^{15}\).
Combining these, we get \(x^{10} y^{15}\).
Understanding the power rule is crucial as it simplifies complex expressions and reduces errors in calculations.
Simplifying expressions
Simplifying expressions is an essential step in solving algebraic problems. It involves reducing an expression to its simplest form to make it easier to work with. Here's how you simplify expressions using the power rule:
  • Identify the terms with exponents.
  • Apply the power rule to each term individually.
  • Combine the simplified terms into one expression.
In the original exercise, we started with \((x^{2} y^{3})^{5}\). By applying the power rule, we simplified it to \(x^{10} y^{15}\).
This process involves straightforward steps, but it's important to follow them carefully to avoid mistakes. Simplifying expressions makes further algebraic manipulations, like addition or multiplication of polynomials, much easier.
Multiplying exponents
When multiplying exponents, you need to pay attention to the rules of exponents. Specifically:
  • When you multiply terms with the same base, add their exponents: \(a^{m} \times a^{n} = a^{m+n}\).
  • When raising an exponent to another exponent, multiply the exponents: \((a^{m})^{n} = a^{m \times n}\).
For the exercise \((x^{2} y^{3})^{5}\), we focused on multiplying the exponents:
  • For \((x^{2})^{5}\), multiplying the exponents gave us \(x^{2 \times 5} = x^{10}\).
  • For \((y^{3})^{5}\), multiplying the exponents gave us \(y^{3 \times 5} = y^{15}\).
This resulted in \(x^{10} y^{15}\).
Mastering these rules helps simplify and solve complex algebraic expressions efficiently.

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Most popular questions from this chapter

Convert each number into scientific notation. $$28.40 \times 10^{6}$$

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 a+5)^{2}$$

In this section we defined \(a^{0}=1 .\) It is important to realize that a definition is neither right nor wrong-it just is. The proper question to ask about a definition is "Is it useful?" What happens if someone decides on an "alternative definition" such as \(a^{0}=7\) because 7 happens to be his or her favorite number? What happens when we consider the expression \(a^{0} \cdot a^{4} ?\) If you use exponent rule 1 you get \(a^{0} \cdot a^{4}=a^{0+4}=a^{4} .\) However, if you use this alternative definition, you get \(a^{0} \cdot a^{4} \stackrel{\underline{2}}{=} 7 a^{4} .\) The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of \(a^{0}=1\) is consistent with all five exponent rules.

Maria can make five flower arrangements per hour, while Francis can make seven flower arrangements per hour. If Maria starts working at 9: 00 A.M. and is joined by Francis at 11: 00 A.M., at what time will they have made 70 flower arrangements?

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you multiply \(1.45 \times 10^{-6}\) by \(2.03 \times 10^{4},\) the result is closest to (a) 10 (b) 100 (c) 1 (d) 0.1 (e) 0.01
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