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Chapter 5: Problem 42

Simplify each expression. $$ \left(\frac{5 t^{8}}{3 m^{3}}\right)^{3} $$

Short Answer

Expert verified
\( \frac{125 t^{24}}{27 m^{9}} \)

Step by step solution

01

Expand the expression

The given expression is \( \left(\frac{5 t^{8}}{3 m^{3}}\right)^{3} \). Begin by applying the power of a quotient rule, which states \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). This gives us \( \frac{(5 t^{8})^{3}}{(3 m^{3})^{3}} \).
02

Simplify the numerator

Now, simplify the numerator \( (5 t^{8})^3 \). Use the power of a product rule \( (ab)^n = a^n b^n \), therefore \( (5 t^{8})^3 = 5^3 (t^{8})^3 = 125 t^{24} \).
03

Simplify the denominator

Simplify the denominator \( (3 m^{3})^3 \) by applying the same power of a product rule: \( (3 m^3)^3 = 3^3 (m^{3})^3 = 27 m^{9} \).
04

Write the simplified expression

Combine the simplified numerator and denominator to write the final expression. Therefore, the simplified form of the initial expression is \( \frac{125 t^{24}}{27 m^{9}} \).

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Key Concepts

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

Exponentiation
Exponentiation is an essential operation in algebra, where a number, called the base, is multiplied by itself a certain number of times, indicated by the exponent. In our given exercise, we are dealing with the exponentiation of a fraction. The overall expression \( \left( \frac{5 t^{8}}{3 m^{3}} \right)^{3} \) exemplifies raising a fraction to the power of 3.
When you raise a fraction to an exponent, you apply the exponent to both the numerator and the denominator separately. This follows the rule:
  • \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \), which means you raise both the top (numerator) and bottom (denominator) of the fraction to the power \( n \).
This property helps simplify complex expressions involving fractions and is fundamental in manipulating expressions for further simplifications.
Fractions in Algebra
Fractions in algebra often represent parts of a whole, but in algebraic terms, they are a way to simplify expressions or denote ratios between variables and constants. Understanding fractions is crucial for operations like simplification, addition, subtraction, and exponentiation of terms.
In our example, fractions play a significant role because we are dealing with an algebraic fraction \( \frac{5 t^{8}}{3 m^{3}} \), where:
  • The numerator \( 5t^8 \) includes a constant and a variable with an exponent.
  • The denominator \( 3m^3 \) also has a constant and a variable with an exponent.
When solving or simplifying expressions that involve fractions, one should pay attention to operations that affect both the numerator and the denominator. Applying the same exponentiation rule to each part simplifies the fraction effectively.
Simplification Steps
Simplification of algebraic expressions involves reducing them into their simplest form. Breaking down the given expression \( \left(\frac{5 t^{8}}{3 m^{3}}\right)^3 \) into smaller, manageable parts is essential. Here's how simplification unfolds:
  • **Simplifying the Numerator**: Start with \( (5 t^{8})^3 \). By following the power of a product rule \( (ab)^n = a^n b^n \), you compute each part separately. First, \( 5^3 = 125 \) and \( (t^{8})^3 = t^{24} \), resulting in the numerator, \( 125 t^{24} \).
  • **Simplifying the Denominator**: Likewise, apply the same principles to \( (3 m^{3})^3 \). Calculate each piece separately: \( 3^3 = 27 \) and \( (m^{3})^3 = m^{9} \), leading to the denominator \( 27 m^{9} \).
By combining these simplified components, you obtain the simplest form of the expression: \( \frac{125 t^{24}}{27 m^{9}} \). This process not only makes algebraic expressions easier to handle but also prevents errors in complex calculations. Simplification ensures clarity, making it a critical step in solving algebraic problems.

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Most popular questions from this chapter

Factor each expression. $$ y^{2}-(2 x-t)^{2} $$

Solve each inequality. Graph the solution set and write it using interval notation. $$ |x|>7 $$

Look Alikes \(. .\) \(*\) $$ \text { a. } 3 y+\left(y^{7}+y\right)+\left(y^{3}+y\right) \text { b. } 3 y\left(y^{7}+y\right)\left(y^{3}+y\right) $$

Solve each inequality. Graph the solution set and write it using interval notation. $$ |2 x+1| \geq 7 $$

Look Alikes... A. \(\quad 3 d^{3}-4 d^{2}-3 d+5\) \(+11 d^{3} \quad-8 d-2\) B. \(\quad 3 d^{3}-4 d^{2}-3 d+5\) \(-\left(11 d^{3} \quad-8 d-2\right)\)
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