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Chapter 6: Problem 159

Simplify each expression. $$\left(10 p^{4}\right)^{3}\left(5 p^{6}\right)^{2}$$

Short Answer

Expert verified
25000p^{24}

Step by step solution

01

Apply the Power Rule

First, apply the power rule to each term. The power rule states that \((a^m)^n = a^{m \times n}\). Apply this rule individually to both \(10p^4\) and \(5p^6\): \[ \left(10p^{4}\right)^{3} = 10^{3} \times p^{4 \times 3} = 10^{3} \times p^{12} \]\[ \left(5p^{6}\right)^{2} = 5^{2} \times p^{6 \times 2} = 5^{2} \times p^{12} \]
02

Calculate the Constants

Now calculate the numerical values of the constants:\[ 10^{3} = 1000 \]\[ 5^{2} = 25 \]
03

Multiply the Constants and Combine the Powers of p

Next, multiply the constants and combine the powers of \(p\):\[ 1000 \times 25 = 25000 \]Combine the like bases by adding the exponents: \[ p^{12 + 12} = p^{24} \]
04

Write the Final Expression

Now, combine the results to write the final simplified expression:\[ 25000 p^{24} \]

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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 is a fundamental concept in algebra when dealing with exponents. It helps in simplifying expressions where a power is raised to another power. The rule states that \((a^m)^n = a^{m \times n}\). This means you multiply the exponents together within each set of parentheses.

For example:
If you have \((x^2)^3\), you use the power rule to multiply the exponents: \[ (x^2)^3 = x^{2 \times 3} = x^6 \].
This makes simplifying complex expressions involving exponents straightforward and manageable. It's an essential tool in algebraic simplification.
Exponents
Exponents are a way to express repeated multiplication of the same number. They are written as a small number above and to the right of the base number.

For instance:
In the term \( 3^4 \), the number 3 is the base, and 4 is the exponent. This means \( 3 \times 3 \times 3 \times 3 = 81 \). Exponents make it easier to write and work with large numbers.

Key points to remember about exponents:
  • Multiplying with the same base: Add the exponents. Example: \( x^2 \times x^3 = x^{2+3} = x^5 \).
  • Dividing with the same base: Subtract the exponents. Example: \( x^5 \div x^3 = x^{5-3} = x^2 \).
  • Raising a power to a power: Multiply the exponents. Example: \( (x^2)^3 = x^{2 \times 3} = x^6 \).
Understanding exponents is critical for simplifying more complex algebraic expressions.
Algebraic Simplification
Algebraic simplification involves reducing expressions to their simplest form. This often includes combining like terms, applying the distributive property, and using rules for exponents.

Here's a step-by-step overview using our example \( \left(10 p^{4}\right)^{3} \left(5 p^{6}\right)^{2} \):
  • Step 1: Apply the power rule separately to each group within the parentheses. This means: \[ (10p^4)^3 = 10^3 \times p^{12} \] and \[ (5p^6)^2 = 5^2 \times p^{12} \].
  • Step 2: Calculate the constants: \[ 10^3 = 1000 \] and \[ 5^2 = 25 \].
  • Step 3: Multiply the constants: \[ 1000 \times 25 = 25000 \]. Then, combine the powers of \( p \) by adding the exponents: \[ p^{12 + 12} = p^{24} \].
  • Step 4: Write the final simplified expression: \[ 25000 p^{24} \].
Algebraic simplification makes complex expressions easy to handle and understand by breaking them down into simpler, manageable parts.

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\frac{72 r^{5} s^{2}+132 r^{4} s^{3}-96 r^{3} s^{5}}{12 r^{2} s^{2}}$$

Simplify. (a) \((-7)^{-2}\) (b) \(-7^{-2}\) (c) \(\left(-\frac{1}{7}\right)^{-2}\) (d) \(-\left(\frac{1}{7}\right)^{-2}\)

Divide each polynomial by the monomial. $$\left(63 a^{2} b^{3}+72 a b^{4}\right) \div(9 a b)$$

Simplify. (a) \(2 \cdot 5^{-1}\) (b) \((2 \cdot 5)^{-1}\)

Divide the monomials. $$\frac{\left(10 m^{5} n^{4}\right)\left(5 m^{3} n^{6}\right)}{25 m^{7} n^{5}}$$
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