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Chapter 12: Problem 114

Simplify. $$ x^{4}\left(x^{5}\right) $$

Short Answer

Expert verified
The simplified expression is \(x^9\).

Step by step solution

01

Identify the Expression

The given expression is \(x^{4}(x^{5})\). This consists of two terms with the same base being multiplied together.
02

Apply the Power Rule for Exponents

When multiplying terms with the same base, you add their exponents. The power rule for exponents states that \( a^m \times a^n = a^{m+n} \).
03

Simplify the Exponents

Add the exponents of the base \(x\) in the expression \(x^{4} \times x^{5}\). This gives \(x^{4+5}\).
04

Calculate the Sum

Add the exponents to get \(x^9\).
05

Write the Simplified Expression

The simplified expression is \(x^{9}\).

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Key Concepts

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

Power Rule for Exponents
In mathematics, the Power Rule for exponents is a helpful guideline that simplifies expressions involving repeated multiplication. When you multiply two terms that have the same base, you can simply add their exponents. This principle is encapsulated in the formula: \(a^m \times a^n = a^{m+n}\). For instance, if you have \(x^4 \times x^5\), you can add the exponents 4 and 5 to get \(x^{4+5}\), which simplifies to \(x^9\). This rule makes calculations easier and more intuitive.
Multiplying Terms with Same Base
When you encounter terms with the same base in a multiplication scenario, there's a straightforward method to simplify your work. Here are the steps:
  • Identify the base in all terms. For example, in \(x^4(x^5)\), the base is \(x\).
  • Apply the Power Rule for Exponents. This means you add the exponents of each term with the same base.
  • Combine the exponents. For \(x^4 \times x^5\), you get \(x^{4+5}\), which simplifies to \(x^9\).
This method works because of the properties of exponents, where multiplying terms with the same base essentially 'stacks' the repeated multiplications.
Adding Exponents
Adding exponents is a crucial part of simplifying expressions involving exponents. When two terms with the same base are multiplied, their exponents are simply added together. Follow these steps to add exponents:
First, ensure the bases are the same. In our example, \(x^4(x^5)\), the base is the same: \(x\).
Next, look at the exponents. Here, the exponents are 4 and 5.
Then, add the exponents together: \(4 + 5 = 9\).
So, \(x^4 \times x^5 \) becomes \(x^9\).
This helps to condense expressions and make calculations much simpler.
Using these strategies can help you feel more confident when confronting exponents in your math work.

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