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Chapter 10: Problem 69

Simplify. $$\left(x^{60} y^{50}\right)^{2}$$

Short Answer

Expert verified
\(x^{120} y^{100}\)

Step by step solution

01

Understand the Expression

The given expression is \((x^{60} y^{50})^2\). It means that \((x^{60} y^{50})\) needs to be squared.
02

Apply the Power Rule

Use the power rule \((a^m)^n = a^{mn}\) for each term inside the parentheses.
03

Simplify Each Term

For \(x^{60}\), applying the power rule gives \((x^{60})^2 = x^{60 \cdot 2} = x^{120}\). Similarly, for \(y^{50}\), \((y^{50})^2 = y^{50 \cdot 2} = y^{100}\).
04

Combine the Results

Combine the simplified terms to get the final result: \(x^{120} y^{100}\).

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Key Concepts

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

Understanding the Power Rule
The power rule is essential when dealing with exponents in algebra. It states that \((a^m)^n = a^{mn}\). This means if you have an exponent raised to another exponent, you multiply the exponents. For example, in our exercise involving \((x^{60} y^{50})^2\), we apply the power rule to each term inside the parentheses.
This rule simplifies the process and helps us manage complex expressions efficiently.
Exponentiation Basics
Exponentiation is a mathematical operation involving numbers called bases and exponents. It represents repeated multiplication. For instance, (\(x^{60}\)) means x is multiplied by itself 60 times.
When you raise an exponentiated term to another power, like \(x^{60}\) raised to 2 in our exercise, you're raising the base to a new power by multiplying the exponents: \(60 \times 2 = 120\).
This results in \(x^{120}\). It's vital to understand this multiplication aspect to simplify expressions effectively.
Working with Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. In our exercise, \((x^{60} y^{50})^2\) is an algebraic expression that needs to be simplified using the power rule.
The two components, \(x^{60}\) and \(y^{50}\), are each raised to the power of 2.
By applying the power rule to each term in the algebraic expression, we simplify the whole expression methodically. Understanding the structure of algebraic expressions is key to applying algebraic rules accurately.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This process includes performing operations and applying algebraic rules like the power rule.
For our given expression \((x^{60} y^{50})^2\), we simplify it by applying the power rule step-by-step:
  • First, \(x^{60}\) raised to 2 gives \((x^{60})^2 = x^{120}\).
  • Second, \(y^{50}\) raised to 2 gives \((y^{50})^2 = y^{100}\).

Combining these results, we get the simplified expression \(x^{120} y^{100}\). Simplification makes expressions easier to understand and work with.

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Most popular questions from this chapter

The cost (in dollars) to rent a car is given by the polynomial \(29.99 x+0.40 y .\) In this context, \(x\) is the number of days that the car is rented and \(y\) is the number of miles driven. a. Evaluate the polynomial for \(x=3\) and \(y=350 .\) Interpret the answer in the context of this problem. b. Evaluate the polynomial for \(x=7\) and \(y=720 .\) Interpret the answer in the context of this problem.

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