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Chapter 10: Problem 31

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$\sqrt{x} \sqrt{x}$$

Short Answer

Expert verified
The simplified expression is \(x\).

Step by step solution

01

- Rewrite the Expression Using Exponents

Rewrite the square root expression in terms of exponents. Recall that \(\text{\( \sqrt{x} \) = x^{\frac{1}{2}} }\).So the given expression becomes \( \sqrt{x} \sqrt{x} = x^{\frac{1}{2}} \cdot x^{\frac{1}{2}} \)
02

- Apply Properties of Exponents

Use the property of exponents which states that when multiplying like bases, you add the exponents: \(\text{\( a^{m} \cdot a^{n} = a^{m+n} \) }\).Thus, \(\text{\( x^{\frac{1}{2}} } \cdot { \frac{1}{2} } \)} = x^{\frac{1}{2} \ + \ \frac{1}{2} } \)
03

- Simplify the Exponents

Add the exponents: \(\text{\( x^{\frac{1}{2}} \cdot x^{\frac{1}{2}} = x^{1} \)} \). Since \(\text{\( x^{1} = x \)} \), the expression simplifies to just \( x \)

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Key Concepts

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

Square Roots
Square roots are one of the foundational mathematical operations, often represented as \( \sqrt{x} \). A square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). The square root is denoted using the radical symbol ( \( \sqrt{} \)) or can be expressed in exponent form as \( x^{\frac{1}{2}} \). Having the expression in this form helps in performing algebraic operations more efficiently since exponent rules are easier to handle than radical symbols.
Exponents
Exponents are a way of expressing repeated multiplication of a number by itself. For example, \( x^3 \) means \( x \times x \times x \). The number \( x \) is called the base, and 3 is the exponent. This notation is compact and useful for large numbers. In algebraic expressions, exponents provide a powerful way to simplify and manipulate terms. To work with square roots, it's useful to remember that \( \sqrt{x} \) is the same as \( x^{\frac{1}{2}} \). This allows you to apply the properties of exponents directly to square roots, making calculations both easier and more consistent.
Properties of Exponents
The properties of exponents are rules that help simplify expressions involving exponents. Some important properties include:
  • Multiplication of like bases: \( a^m \cdot a^n = a^{m+n} \)
  • Division of like bases: \( \frac{a^m}{a^n} = a^{m-n} \)
  • Power of a power: \( (a^m)^n = a^{m \cdot n} \)
  • Product to a power: \( (ab)^m = a^m \cdot b^m \)
  • Quotient to a power: \( \( \frac{a}{b} \) ^n = \frac{a^n}{b^n} \)
These properties allow for the simplification and evaluation of expressions like \( \sqrt{x} \sqrt{x} \). Here, rewriting the square root in exponent form ( \( x^{\frac{1}{2}} \)) and using the multiplication rule (adding exponents) lets us combine the terms easily, eventually simplifying to \( x \).

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Most popular questions from this chapter

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$4 \sqrt{96}-5 \sqrt{24}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{18}+\sqrt{27}+\sqrt{45}$$

Sketch the graph of each equation in a rectangular coordinate system. Label the intercepts. $$5 y-2 x=10$$

Solve the following problems algebraically. Be sure to label what the variable represents. Xavier made three investments at \(6.5 \%, 7.6 \%,\) and \(9.2 \% .\) The amount invested at \(7.6 \%\) is \(\$ 1000\) less than the amount invested at \(9.2 \%,\) and the amount invested at \(6.5 \%\) is twice the amount invested at \(7.6 \% .\) If the annual income from the three investments is \(\$ 837,\) how much is invested altogether?

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$b^{2} \sqrt{40 a^{5} b^{2}}+a \sqrt{90 a^{3} b^{6}}$$
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