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Chapter 7: Problem 73

Solve each equation. $$(3 a-1)^{-2 / 5}=1$$

Short Answer

Expert verified
The solution is \( a = \frac{2}{3} \).

Step by step solution

01

Understand the Equation

The given equation is \( (3a - 1)^{-2/5} = 1 \). The goal is to solve for \( a \) by isolating it.
02

Isolate the Base

To eliminate the exponent \( -2/5 \), recognize that any number to the power of zero is 1. Thus, \( (3a - 1)^{-2/5} = 1 \) implies \( 3a - 1 \) must be 1.
03

Set the Base Equal to 1

Since any base raised to a power of zero equals 1, we can set the base equal to 1: \( 3a - 1 = 1 \).
04

Solve for a

Add 1 to both sides: \( 3a - 1 + 1 = 1 + 1 \) which simplifies to \( 3a = 2 \).
05

Divide Both Sides

Finally, divide both sides by 3: \( a = \frac{2}{3} \). This is the solution.

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Key Concepts

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

algebra
Algebra is the branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It involves operations such as addition, subtraction, multiplication, and division to solve for unknown variables. The key to mastering algebra is understanding the rules and properties of these operations. For example:
  • Combining like terms in an equation simplifies it.
  • The distributive property allows you to expand and simplify expressions.
  • To solve for a variable, you can perform the same operation on both sides of the equation.
By following these principles, algebra becomes a powerful tool for solving various mathematical problems.
exponents
Exponents are a way of expressing repeated multiplication of the same number. For instance, \(2^3\) means \(2 \times 2 \times 2\), which equals 8. Exponents have specific rules that make calculations easier:
  • Any number raised to the power of zero is 1, for example, \(5^0 = 1\).
  • The power of a power rule states that \((a^m)^n = a^{mn}\).
  • The product of powers rule tells us that \(a^m \times a^n = a^{m+n}\).
  • The quotient of powers rule indicates \(a^m / a^n = a^{m-n}\).
In the given exercise, understanding these rules simplifies the process of solving for the variable.
solving for variables

Solving for variables involves isolating the unknown variable on one side of the equation. This process often includes:
  • Performing inverse operations to cancel out terms.
  • Simplifying expressions by combining like terms.
  • Ensuring the steps taken are legal by doing the same operation to both sides of the equation.
In the exercise, we started with the equation \((3a - 1)^{-2/5} = 1\). Using exponent rules, we recognized that any number raised to zero is 1. This allowed us to set the base equal to 1. By isolating the variable step-by-step, we ultimately found that \(a = \frac{2}{3}\). This methodical approach is crucial for solving equations correctly.

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