Problem 10 - Elementary and Intermediate Algebra

Question

Determine whether each statement is true or false. \(\begin{array}{ll}{\text { a. } 6^{-2}=-36} & {\text { b. } 6^{-2}=\frac{1}{36}} \\ {\text { c. } \frac{x^{3}}{y^{-2}}=\frac{y^{2}}{x^{3}}} & {\text { d. } \frac{-6 x^{-5}}{y^{-6}}=\frac{y^{6}}{6 x^{5}}}\end{array}\)

Step-by-Step Solution

Answer

a: False, b: True, c: True, d: False.

Step 1: Evaluate statement a

Statement a is \( 6^{-2} = -36 \). To evaluate this, recall that \( a^{-n} = \frac{1}{a^n} \). Therefore, \( 6^{-2} \) is \( \frac{1}{6^2} = \frac{1}{36} \), not \(-36\). The statement is false.

Step 2: Evaluate statement b

Statement b is \( 6^{-2} = \frac{1}{36} \). Using the property \( a^{-n} = \frac{1}{a^n} \), we find that \( 6^{-2} = \frac{1}{6^2} = \frac{1}{36} \). The statement is true.

Step 3: Evaluate statement c

Statement c is \( \frac{x^{3}}{y^{-2}}=\frac{y^{2}}{x^{3}} \). To simplify the left side, recall that \( y^{-n} = \frac{1}{y^n} \), so \( \frac{x^3}{y^{-2}} = x^3 \times y^2 = \frac{y^2 x^3}{1} \). Clearly, this matches the right side when simplified. Therefore, the statement is true.

Step 4: Evaluate statement d

Statement d is \( \frac{-6 x^{-5}}{y^{-6}}=\frac{y^{6}}{6 x^{5}} \). By rewriting negative exponents, \( x^{-5} \) becomes \( \frac{1}{x^5} \) and \( y^{-6} \) becomes \( \frac{1}{y^6} \). Hence, \( \frac{-6 x^{-5}}{y^{-6}} = -6 \times \frac{y^6}{x^5} \). The statement simplifies to \( \frac{-6 y^6}{x^5} \), which is not equal to \( \frac{y^6}{6 x^5} \). Therefore, the statement is false.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential to evaluate algebraic expressions accurately. An exponent indicates how many times a number, known as the base, is multiplied by itself. A key rule is the negative exponent property, which states that any number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent.

• If you have a number like \( a^{-n} \), it transforms to \( \frac{1}{a^n} \).
• This means \( 6^{-2} \) becomes \( \frac{1}{6^2} = \frac{1}{36} \).
• Expressions like \( x^{-5} \) become \( \frac{1}{x^5} \).

These guidelines help in converting and simplifying expressions with negative exponents.

Simplifying Expressions

Simplifying algebraic expressions involves transforming them into a form that is easier to work with. This often includes applying exponent rules and performing basic arithmetic operations.

• The expression \( \frac{x^3}{y^{-2}} \) can be simplified by changing \( y^{-2} \) to \( y^2 \), resulting in \( x^3 \times y^2 \).
• For expressions like \( \frac{-6x^{-5}}{y^{-6}} \), rewrite \( x^{-5} \) as \( \frac{1}{x^5} \) and \( y^{-6} \) as \( y^6 \), then simplify.
• Ensure each term is rewritten with positive exponents to avoid mistakes.

Simplifying correctly allows for comparing and combining expressions accurately, as seen when verifying expressions in the original exercise.

True or False Statements

True or false exercises in algebra test your understanding of fundamental math concepts, such as exponent rules and simplifying expressions. They require careful evaluation to ensure accuracy.

• To verify a statement like \( 6^{-2} = -36 \), convert the exponent and calculate to find it is actually \( \frac{1}{36} \), making the statement false.
• For \( \frac{x^3}{y^{-2}} = \frac{y^2}{x^3} \), simplify both sides of the equation to confirm they match, proving it true.
• Evaluating statements requires checking each step for accuracy in conversion and simplification.

Through practice, you develop an intuitive sense for assessing the validity of such algebraic statements, enhancing problem-solving skills.