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Chapter 10: Problem 103

In the following exercises, simplify each expression. $$\left(x^{2}\right)^{4} \cdot\left(x^{3}\right)^{2}$$

Short Answer

Expert verified
The expression simplifies to \(x^{14}\).

Step by step solution

01

Apply the Power Rule

Use the power rule \((a^m)^n = a^{mn}\) to simplify each term separately. For \(x^2\) raised to the 4th power, apply the rule: \((x^2)^4 = x^{2 \times 4} = x^8\). Similarly, for \(x^3\) raised to the 2nd power, apply the rule: \((x^3)^2 = x^{3 \times 2} = x^6\).
02

Multiply the Results

Now, multiply the simplified results from step 1. You have \(x^8 \cdot x^6\).
03

Apply the Product Rule

Use the product rule \((a^m \cdot a^n = a^{m+n})\) to combine the exponents. For \(x^8 \cdot x^6\), the exponents are added: \(x^{8+6} = x^{14}\).

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Key Concepts

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

Power Rule
The power rule in algebra helps us simplify expressions where a term with an exponent is raised to another exponent. The rule states: ewline \((a^m)^n = a^{m*n}\). ewline For example, if you have an expression like \((x^2)^4\), here's what you do:ewline
  • Identify the base \(x\) and the exponents \(2\) and \(4\).
  • Multiply the exponents together: \(2 \times 4 = 8\).
  • The simplified form is \(x^8\).
Using this rule makes expressions much easier to handle. By breaking it down, you simplify each part before combining them together into one simpler expression.
Product Rule
To combine terms with the same base raised to exponents, we use the product rule. This rule states: ewline \((a^m \cdot a^n = a^{m+n})\). ewline If you have an expression like \(x^8 \cdot x^6\), here's how to simplify it:ewline
  • Ensure that the bases are the same - in this case, both are \(x\).
  • Add the exponents: \(8 + 6 = 14\).
  • The simplified form is \(x^{14}\).
The product rule helps in combining terms efficiently, reducing complex algebraic expressions to simpler forms without changing their values. This rule is crucial whenever you're working with multiplication of like bases with different exponents.
Algebraic Exponents
Exponents in algebra signify the number of times a base is multiplied by itself. Understanding how they work is key to simplifying expressions.ewline
  • In the power rule, you raise a term with an exponent to another exponent. This effectively 鈥榩owers up鈥� the term even further.
  • In the product rule, you combine like terms by adding their exponents when multiplying.
  • The rules for exponents make handling large numbers manageable by breaking them down into simpler steps.
It's important to grasp these fundamentals as they are the building blocks for more advanced algebraic concepts. With practice, using exponents becomes intuitive, making complex algebra equations easier to solve and understand.

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