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Chapter 9: Problem 53

Simplify. $$\left(x^{7}\right)^{2}$$

Short Answer

Expert verified
The simplified form of \(\left(x^{7}\right)^{2}\) is \(x^{14}\).

Step by step solution

01

Identify the problem

The given problem is \(\left(x^{7}\right)^{2}\), which is a power of a power problem. In such a problem, you will use the rule that \((a^m)^n = a^{mn}\).
02

Apply the power of a power rule

According to the rule, this simplifies to \(x^{(7)(2)}\).
03

Calculate the exponent

Calculate the product of 7 and 2, getting \(x^{14}\).

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental concept in mathematics, acting as a shortcut for multiplication. When you see something like \( x^7 \), it means \( x \) multiplied by itself seven times. This concept extends to any whole number, fractional, or even negative exponents, each representing different operations on numbers or variables.

Think about everyday examples to make this abstract concept more tangible:
  • Scientific notations often use exponents, like \( 3 \times 10^2 \), making it easier to express large numbers.
  • Growth phenomena, such as population or compound interest, are often described using exponents.
Grasping exponents is key to tackling more complex operations in algebra and understanding how they interact in different mathematical contexts.
Simplifying Expressions Using Power of a Power Rule
To simplify expressions like \( \left(x^{7}\right)^{2} \), we apply the power of a power rule. This rule states that \( (a^m)^n = a^{mn} \). It's like folding one operation into another for a more straightforward calculation.

Let鈥檚 walk through the steps for our example:
  • Identify that the expression is \( \left(x^{7}\right)^{2} \).
  • Recognize that it's a power raised to another power, triggering the use of the power of a power rule.
  • Apply the rule: \( x^{7 \cdot 2} \).
  • Finally, simplify the expression to \( x^{14} \).
This method not only makes the expression more manageable but also clarifies the relationships in abstract problems.

Practicing this rule regularly can make it a second nature skill, especially useful in algebraic manipulation and higher-level mathematics.
The Importance of Mathematics Education in Mastering Exponents
Mathematics education plays a crucial role in helping students understand intricate concepts such as exponents. It provides a structured way to explore and internalize these ideas, offering both the tools and the context needed to navigate mathematical challenges.

There are several ways through which math education makes learning about exponents effective:
  • Use of visual aids: Diagrams and visualizations help students see patterns and concepts, making it easier to grasp exponent rules.
  • Hands-on activities: Engaging in practical exercises fosters a deeper understanding and retention of material.
  • Real-world applications: Showing how exponents function in real life鈥攍ike in technology and sciences鈥攂ridges the gap between theory and practical use.
Through comprehensive education, learners are encouraged not only to memorize rules but also to understand their derivations and applications.

This understanding is vital in building a strong foundation for future studies and everyday problem-solving.

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Most popular questions from this chapter

The radius of a circle is \((x+4) \mathrm{cm} .\) Find the area of the circle in terms of the variable \(x\). Leave the answer in terms of \(\pi\).

Complete. If \(m=n\) and \(a \neq 0,\) then \(\frac{a^{m}}{a^{a}}=\text{_____}\)

Expand. $$(2 a-9 b)^{2}$$

For Exercises, write in decimal notation. $$2 \times 10^{15}$$

State whether the coefficient of the \(x\) term of the product is positive, negative, or zero. $$(a x+b)^{2}, \text { where } a>0 \text { and } b>0$$
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