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Chapter 1: Problem 66

Simplify. $$ \left(x^{4} \cdot x^{3}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \(x^{14}\).

Step by step solution

01

Apply the Power Rule for Exponents

First, simplify the expression inside the parentheses by using the power rule for exponents, which states that when multiplying like bases, you add the exponents. In this case, the like bases are both 'x'. So add the exponents 4 and 3: \(x^4 \cdot x^3 = x^{4+3} = x^7\).
02

Apply the Power of a Power Rule

Now apply the power of a power rule to the entire expression \((x^7)^2\),which means you multiply the exponents: \((x^7)^2 = x^{7 \cdot 2} = x^{14}\).
03

Write the Final Simplified Expression

After applying the rules, you arrive at the final simplified expression. Therefore, \((x^4 \cdot x^3)^2\) simplifies to \(x^{14}\).

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Key Concepts

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

Power Rule for Exponents
The power rule for exponents is an essential concept when dealing with expressions that have like bases. It states that when you multiply two algebraic terms with the same base, you simply add their exponents. This is symbolically represented as:
  • \(a^m\cdot a^n = a^{m+n}\)
In our example, we have \(x^4\) and \(x^3\) being multiplied inside the parentheses. Both these terms share the same base 'x'. So, according to the power rule for exponents, we add the exponents 4 and 3. This simplifies to:
  • \(x^4 \cdot x^3 = x^{4+3} = x^7\)
Using this method, we reduce the complexity by focusing solely on the exponents, making it easier to handle larger terms. Remember, this rule only applies when the bases are identical.
Power of a Power Rule
The power of a power rule comes into play when you take an exponential expression and raise it to another power. This rule dictates that you multiply the exponents. The general form is:
  • \((a^m)^n = a^{m\cdot n}\)
In our exercise, after simplifying the inner pair \((x^4 \cdot x^3)\) to \(x^7\), we then raise this term to the power of 2. Applying the power of a power rule means multiplying the exponent of the expression by the external power:
  • \((x^7)^2 = x^{7 \cdot 2} = x^{14}\)
This step is crucial for condensing expressions further and helps maintain order when faced with nested exponentials.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form, making them easier to read and work with. The process requires understanding and applying the rules of exponents correctly to consolidate terms. In our scenario, we started with a more complex expression:
  • \((x^4 \cdot x^3)^2\)
By using the rules for exponents, we transformed this expression into a much simpler form, \(x^{14}\). Simplifying like this:
  • Combines like terms, reducing clutter.
  • Makes subsequent mathematical operations more manageable.
  • Helps to clearly identify core terms and coefficients.
A simplified expression not only looks cleaner but also allows for quicker problem solving in various contexts of mathematics.

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Most popular questions from this chapter

world population (in millions) since the year 1700 is approximated by the exponential function \(P(x)=522(1.0053)^{x}\), where \(x\) is the number of years since 1700 (for \(0 \leq x \leq 200\) ). Using a calculator, esti mate the world population in the year: 1750

ECONOMICS: Per Capita Personal Income In the short run, per capita personal income (PCPI) in the United States grows approximately linearly. In 2001 PCPI was \(30.4\), and in 2009 it had grown to \(39.2\) (both in thousands of dollars). a. Use the two given (year, PCPI) data points \((1,30.4)\) and \((9,39.2)\) to find the linear relationship \(y=m x+b\) between \(x=\) years since 2000 and \(y=\mathrm{PCPI}\). b. Use your linear relationship to predict PCPI in 2020 .

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