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Chapter 7: Problem 25

Write each expression in exponential form. $$\sqrt[3]{(5 x y)^{6}}$$

Short Answer

Expert verified
The expression \( \sqrt[3]{(5 x y)^{6}} \) in exponential form is \( 25x^{2}y^{2} \).

Step by step solution

01

Represent Roots as Exponents

Radicals (or roots) can be expressed as fractional exponents. For instance, a cube root is equivalent to an exponent of \( \frac{1}{3} \). Therefore, \( \sqrt[3]{(5 x y)^{6}} \) can be expressed as \( (5xy)^{\frac{6}{3}} \).
02

Simplify the Exponent

Now simplify the exponent. Divide 6 by 3. This results in \( (5xy)^{2} \).
03

Distribute the Exponent

The last step is distributing the exponent to each term in the brackets. Hence we get \( 5^{2} * x^{2} * y^{2} \) which simplifies to \( 25x^{2}y^{2} \).

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Key Concepts

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

Fractional Exponents
Fractional exponents are a way to express roots in exponential form. They make working with roots simpler and allow us to use the same rules for both powers and roots. A fractional exponent is expressed as a number in the form of a fraction, where the numerator is the power and the denominator is the root. For example, the cube root of a number, such as \( \sqrt[3]{a} \), is written as \( a^{\frac{1}{3}} \). This concept allows us to easily convert between radicals and power notation, simplifying our calculations. Remember: the larger the denominator, the smaller the fractional exponent, meaning a weaker root operation.
Exponent Rules
When working with expressions in exponential form, there are several key exponent rules that can help simplify expressions. One fundamental rule is the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\). This means you'll multiply the exponents together when a power is raised to another power. Another essential rule is the product of powers rule, which states that \(a^m \cdot a^n = a^{m+n}\). This allows you to add the exponents when multiplying the same bases. Lastly, when dividing like bases, you use the quotient of powers rule: \(a^m / a^n = a^{m-n}\). These rules are crucial when transforming and simplifying expressions involving exponents.
Cube Roots
Cube roots are a specific type of radical expression that can be easily expressed using fractional exponents. The cube root of a number \(a\) is the value \(x\) such that \(x^3 = a\). This radical form can be converted to an exponential form as \(a^{\frac{1}{3}}\). So, when we see an expression under a cube root, such as \( \sqrt[3]{(5xy)^6} \), we can convert it to \( (5xy)^{\frac{6}{3}} \), which makes it easier to simplify further. Cube roots are pivotal in rewiring our understanding of radicals within exponential expressions, aiding in the clearer application of exponent laws.
Simplifying Expressions
Simplifying expressions involves making them as concise as possible while maintaining their original value. This means reducing expressions to their simplest form, often by applying exponent rules or converting roots to fractional exponents. For example, simplifying \( (5xy)^{\frac{6}{3}} \) involves first simplifying the exponent \(\frac{6}{3}\), which equals 2. Hence, the expression becomes \( (5xy)^{2} \). Distributing the power further results in \(5^{2} \cdot x^{2} \cdot y^{2}\), or \(25x^{2}y^{2}\). This process involves attention to detail and understanding of mathematical operations, ensuring that complex expressions are always reduced to their simplest form.

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Most popular questions from this chapter

What is the inverse of \(y=5 x-1 ?\) $$ \begin{array}{lllll}{\text { A. } y=5 x+1} & {\text { B. } y=\frac{x+1}{5}} & {\text { C. } y=\frac{x}{5}+1} & {\text { D. } y=\frac{x}{5}-1}\end{array} $$

Graph each function. \(y=\sqrt[3]{x+5}\)

The size of a television screen is the length of the screen's diagonal \(d\) in inches. The equation \(d=\sqrt{2 A}\) models the length of a diagonal of a television screen with area \(A .\) a. Graph the equation on your calculator. b. Suppose you want to buy a new television that has twice the area of your old television. Your old television has an area of 100 in. \(2 .\) What size screen should you buy?

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(\sqrt{2 x+5}=\sqrt{2-x}\)

Find the inverse of each function. Is the inverse a function? \(f(x)=\frac{2}{3} x-3\)
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