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Chapter 10: Problem 31

$$\text { Fill in the blanks }\left(x^{3}\right)^{5}=x \square \cdot \square=x^{15}$$

Short Answer

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3; 5

Step by step solution

01

Apply the Power Rule of Exponents

When raising a power to another power, you multiply the exponents. The rule is \((a^m)^n = a^{m \times n}\). For the given problem, \(x^3\) raised to the power of 5 can be written as \(x^{3 \times 5}\).
02

Multiply the Exponents

Multiply the exponents: \(3 \times 5 = 15\). Thus, \( \left(x^{3}\right)^{5} = x^{3 \times 5} = x^{15}\).
03

Fill in the Blanks

Identify what values go in the blanks. It becomes clear that \((x^{3})^{5} = x^{15}\) without needing additional multiplication. Therefore, \ x^{15} \ is already the simplified form.

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Key Concepts

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

Exponentiation
Exponentiation is a fundamental mathematical operation that involves raising a base number to a given power. The power, also known as an exponent, indicates how many times the base number is multiplied by itself. For instance, in the expression \(x^3\), the base is \(x\) and the exponent is 3, meaning that \(x\) is multiplied by itself three times: \(x \times x \times x\).
Common properties of exponentiation include:
  • \(a^0 = 1\) for any nonzero number \(a\).
  • \(a^1 = a\).
  • \(a^m \times a^n = a^{m+n}\).
These rules help simplify expressions involving exponents.
Multiplication of Exponents
Multiplying exponents occurs when you have a power raised to another power, such as in \( (a^m)^n \). This is where the Power Rule of Exponents comes into play. According to the Power Rule: \( (a^m)^n = a^{m \times n} \).
In the given exercise, we need to simplify \((x^3)^5\). To do this, we multiply the exponents: \(3\) and \(5\). So:
\( (x^3)^5 = x^{3 \times 5} = x^{15} \).
This rule is crucial for working with more complex exponential expressions and ensures that calculations are efficient and accurate.
Simplifying Exponents
Simplifying exponents involves rewriting expressions in their most reduced form. Using the Power Rule, you can often combine or reduce terms for easier computation.
Let's look at our example:
\( (x^3)^5 = x^{15} \)
This expression is already simplified because we applied the Power Rule correctly. There are no further reductions needed because \(x^{15}\) is the most simplified way to express the result.
Key steps to remember when simplifying exponents include:
  • Identify and apply the correct exponent rules (such as Power Rule, Product Rule, etc.).
  • Perform necessary calculations, like multiplying exponents.
  • Ensure the expression is in its simplest form.
By following these steps, you can handle even the most complex exponential expressions with confidence.

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