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Chapter 0: Problem 98

Simplify using properties of exponents. $$ \left(125 x^{9} y^{6}\right)^{\frac{1}{3}} $$

Short Answer

Expert verified
The simplified form of \( (125x^{9}y^{6})^{1/3} \) is \( 5x^{3}y^{2} \).

Step by step solution

01

Express the Problem in Radicals

Let's first understand the given expression \( (125 x^{9} y^{6})^{1/3} \) in terms of radicals. The expression means the cubic root of \(125 x^{9} y^{6}\). The cubic root is written as \(\sqrt[3]{125 x^{9} y^{6}}\).
02

Break down the terms

Since multiplication is a commutative operation, the root of multiplied terms can be broken down into multiplied roots of individual terms. Accordingly, we can rewrite the above expression as \( \sqrt[3]{125} \cdot \sqrt[3]{x^{9}} \cdot \sqrt[3]{ y^{6}} \)
03

Compute Cube roots

Now, we compute the cube roots. The cube root of 125 is 5, since \(5^{3} = 125\). For the variables, since they are both raised to an exponent and inside a root, we apply the rule that states 'when raising an exponent to another exponent, you multiply these exponents'. Thus, the cube root of \(x^{9}\) is \(x^{3}\) and the cube root of \(y^{6}\) is \(y^{2}\).
04

Combine the Results

Bringing all the computed cube roots together, the simplified form of the given expression is \( 5x^{3}y^{2}\).

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Key Concepts

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

Cubic Roots
Cubic roots can be a bit tricky to grasp at first, but they are simply the opposite of cubing a number. When we talk about the cubic root, we're looking for a number which, when multiplied by itself three times, gives the original number.
For example, when we say \(\sqrt[3]{8}\), we're looking for the number that makes \(x^3 = 8\). That number is 2 because \(2 imes 2 imes 2 = 8\).
In the original problem, we are tasked with finding \(\sqrt[3]{125}\). Here, 5 is the cubic root because \(5 imes 5 imes 5 = 125\).
Cubic roots play an essential role in simplifying algebraic expressions, allowing us to break down complex terms into more manageable components.
Simplifying Expressions
Simplifying expressions involves breaking them down into a simpler or more compact form without changing their value. This usually makes them easier to work with and understand.
In the context of the original exercise, the expression involves a combination of numbers and variables raised to powers. To simplify, we first interpret the expression as a cubic root and then break it down step-by-step.
Here's how:
  • The numerical part \(125\) is reduced to its cubic root, which is 5.
  • The expression \(\sqrt[3]{x^9}\) simplifies to \(x^{3}\).Because we're applying the exponent rule of multiplying exponents, which we'll cover shortly.
  • Similarly, \(\sqrt[3]{y^6}\) simplifies to \(y^{2}\).
Combining the simplified parts gives us the expression \(5x^{3}y^{2}\).
Exponent Rules
Exponent rules are crucial in algebra for simplifying expressions and solving equations. One key rule is the 鈥榩ower of a power鈥� rule, which you'll find particularly useful when dealing with roots and powers at the same time.
When you have an exponent raised to another exponent, you multiply the exponents. Mathematically, this means: \((a^{m})^{n} = a^{m \\) \}\").For example, \(x^9)^{1/3}\) converts to \(x^{9 \ rac{1}{3}}\), which simplifies to \(x^3\).
This rule also applies to y's expression: \(y^6)^{1/3}\) becomes \(y^{2}\). By consistently applying exponent rules, expressions become more straightforward, facilitating easier problem-solving in algebra.

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Most popular questions from this chapter

In one short sentence, five words or less, explain what $$ \frac{\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}}{\frac{1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}} $$ does to each number \(x\).

Factor Completely. $$x^{4}-5 x^{2} y^{2}+4 y^{4}$$

What is an algebraic expression? Give an example with your explanation.

will help you prepare for the material covered in the first section of the next chapter. If \(y=1-x^{2}\), find the value of \(y\) that corresponds to values of \(x\) for each integer starting with \(-3\) and ending with 3 .

Factor Completely. $$(y+1)^{3}+1$$
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