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Chapter 10: Problem 63

Simplify. $$ \left(\frac{1}{3} m\right)^{2} $$

Short Answer

Expert verified
The simplified form of \((\frac{1}{3}m)^2\) is \(\frac{1}{9}m^2\).

Step by step solution

01

Apply the power

To simplify \((\frac{1}{3}m)^2\), apply the power of 2 to both the fraction \(\frac{1}{3}\) and the variable \(m\). So, \((\frac{1}{3}m)^2= (\frac{1}{3})^2 \cdot m^2\).
02

Calculate the Square of the Fraction

The square of \(\frac{1}{3}\) is \(\frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\). So, substitute this value back into the expression: \((\frac{1}{3}m)^2= (\frac{1}{3})^2 \cdot m^2 = \frac{1}{9} m^2\).
03

Validate the Expression

Finally, verify whether there are no further steps to simplify the expression. In this case, \(\frac{1}{9}m^2\) is the simplest form and cannot be reduced.

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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 foundational concept in algebra, representing repeated multiplication of a number by itself. When you see an expression like \(a^b\), it reads as "\(a\) raised to the power of \(b\)." This means you multiply \(a\) by itself \(b\) times.

In this exercise, the exponent 2 indicates that you multiply the base, which is \((\frac{1}{3}m)\), by itself: \((\frac{1}{3}m) \times (\frac{1}{3}m)\).
  • Whenever you see a number or variable with an exponent, remember that the exponent tells you how many times to use the base in a multiplication.鈥�
  • Always apply the exponent to both the numerical and variable parts separately, like \((\frac{1}{3})^2\) and \(m^2\).
  • Practicing with exponents helps in solving larger algebra problems and understanding scientific notation.
Recognizing how exponents work will make simplification tasks more intuitive and manageable.
Dealing with Fractions
Fractions represent parts of a whole and are written in the form \(\frac{numerator}{denominator}\). Simplifying fractions involves numerous algebraic manipulations.

In our example, \(\frac{1}{3}\) becomes smaller when squared, specifically \((\frac{1}{3})^2 = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\). This shows how multiplying fractional exponents can lead to even smaller numbers.
  • Always square the numerator and the denominator separately.
  • Understanding how to manipulate fractions is crucial in algebra.
  • Getting comfortable with fractions will help you handle more complex equations involving division and ratios.
Knowing these basic operations with fractions can significantly simplify complex algebraic expressions.
Simplification of Polynomial Expressions
Polynomial expressions consist of variables and coefficients. They are often involved in algebraic manipulations and simplifications.

In this problem, the expression involves multiplying \(m\) by itself, resulting in \(m^2\). When combined with our fraction, it becomes \(\frac{1}{9}m^2\).
  • Simplifying expressions involves reducing them to their smallest form by applying algebra rules.
  • This particular expression is a simple polynomial where there are no additional terms to simplify further.
  • Understanding polynomial expressions is vital when working with equations and functions in advanced math.
Simplifying polynomials allows you to solve equations quickly and understand the core structure of algebraic expressions.

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Most popular questions from this chapter

Factor the expression. Tell which special product factoring pattern you used. $$169-x^{2}$$

Use the following information. In the sport of pole-vaulting, the height \(h\) (in feet) reached by a pole- vaulter is a function of \(v,\) the velocity of the pole-vaulter, as shown in the model below. The constant \(g\) is approximately 32 feet per second per second. Pole-vaulter height model: \(h=\frac{v^{2}}{2 g}\) To reach a height of 16 feet, what is the pole-vaulter's velocity?

Which one of the following is a correct factorization of the expression \(-16 x^{2}+36 x+52 ?\). A) \((-16 x+26)(x-2)\) (B) \((-4 x-13)(4 x+4)\) (C) \(-1(4 x-13)(4 x+4)\) (D) \(-1(4 x-4)(4 x+13)\)

Which of the following is the solution of the equation \(-4 x^{2}+24 x-36=0 ?\) A) -6 B) -3 C) 2 D) 3

Simplify the expression. $$\sqrt{4} \cdot 3 \sqrt{9}$$
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