Mastering the Quotient Rule for Exponents

Q1. Simplify the expression: $\frac{y^7}{y^5}$

Background

Topic: Quotient Rule for Exponents

This question tests your understanding of how to simplify expressions with exponents when dividing terms with the same base.

Key Terms and Formulas

• Quotient Rule for Exponents: When dividing like bases, subtract the exponents.

• $\frac{a^m}{a^n} = a^{m-n}$

Step-by-Step Guidance

1. Identify the base in both the numerator and denominator. Here, the base is $y$.

2. Check the exponents: The numerator has $y^7$ and the denominator has $y^5$.

3. Apply the quotient rule: Subtract the exponent in the denominator from the exponent in the numerator.

4. Set up the subtraction: $7 - 5$.

Try solving on your own before revealing the answer!

Final Answer: $y^2$

Using the quotient rule, $\frac{y^7}{y^5} = y^{7-5} = y^2$.

This means you subtract the exponents because the bases are the same.

Q2. Simplify the expression: $\frac{m^6}{m^6}$

Background

Topic: Zero Exponent Rule and Quotient Rule for Exponents

This question tests your understanding of what happens when you divide a term by itself, especially with exponents.

Key Terms and Formulas

• Quotient Rule for Exponents: $\frac{a^m}{a^n} = a^{m-n}$

• Zero Exponent Rule: $a^0 = 1$ (for $a \neq 0$)

Step-by-Step Guidance

1. Notice that the numerator and denominator have the same base and exponent: $m^6$.

2. Apply the quotient rule: Subtract the exponents in the denominator from the numerator.

3. Set up the subtraction: $6 - 6$.

4. Recall the zero exponent rule: Any nonzero number to the zero power equals 1.

Try solving on your own before revealing the answer!

Final Answer: $1$

$\frac{m^6}{m^6} = m^{6-6} = m^0 = 1$ (as long as $m \neq 0$).

Any nonzero number raised to the zero power is 1.