College Algebra Study Guide: Multiplying, Dividing, and Simplifying Exponents

Q1. Multiply and simplify: $3x^2 \cdot 6x^4$

Background

Topic: Properties of Exponents

This question tests your ability to multiply monomials and apply the rules for exponents.

Key Terms and Formulas:

• Product Rule for Exponents: $a^m \cdot a^n = a^{m+n}$

• Multiplying coefficients: Multiply the numbers in front of the variables.

Step-by-Step Guidance

1. Multiply the coefficients: $3 \cdot 6$.

2. Apply the product rule to the exponents: $x^2 \cdot x^4 = x^{2+4}$.

3. Combine your results to write the expression in simplified form.

Try solving on your own before revealing the answer!

Final Answer: $18x^6$

We multiplied the coefficients and added the exponents according to the product rule.

Q2. Divide and simplify: $8x^7 \div 2x^3$

Background

Topic: Properties of Exponents

This question tests your ability to divide monomials and apply the quotient rule for exponents.

Key Terms and Formulas:

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

• Divide coefficients: $8 \div 2$

Step-by-Step Guidance

1. Divide the coefficients: $8 \div 2$.

2. Apply the quotient rule to the exponents: $x^7 \div x^3 = x^{7-3}$.

3. Combine your results to write the expression in simplified form.

Try solving on your own before revealing the answer!

Final Answer: $4x^4$

We divided the coefficients and subtracted the exponents according to the quotient rule.

Q3. Simplify: $(6a^2b^3c)^2$

Background

Topic: Properties of Exponents

This question tests your ability to apply the power rule to a product of terms.

Key Terms and Formulas:

• Power Rule: $(a^m)^n = a^{mn}$

• Distribute the exponent to each factor inside the parentheses.

Step-by-Step Guidance

1. Apply the exponent to the coefficient: $(6)^2$.

2. Apply the power rule to each variable: $(a^2)^2 = a^{2 \cdot 2}$, $(b^3)^2 = b^{3 \cdot 2}$, $(c)^2 = c^{1 \cdot 2}$.

3. Combine all the results to write the expression in simplified form.

Try solving on your own before revealing the answer!

Final Answer: $36a^4b^6c^2$

We squared the coefficient and multiplied the exponents for each variable.

Q4. Simplify: $\frac{a^7b^2}{a^3b^5}$

Background

Topic: Properties of Exponents

This question tests your ability to simplify a fraction using the quotient rule for exponents.

Key Terms and Formulas:

• Quotient Rule: $a^m \div a^n = a^{m-n}$

• Apply the rule separately to each variable.

Step-by-Step Guidance

1. Apply the quotient rule to $a$: $a^7 \div a^3 = a^{7-3}$.

2. Apply the quotient rule to $b$: $b^2 \div b^5 = b^{2-5}$.

3. Combine the results to write the simplified expression.

Try solving on your own before revealing the answer!

Final Answer: $a^4b^{-3}$ or $\frac{a^4}{b^3}$

We subtracted the exponents for each variable. Negative exponents indicate reciprocals.

Q5. Simplify: $\frac{2x^3y^2}{3x^2y^5}$

Background

Topic: Properties of Exponents

This question tests your ability to simplify a fraction with multiple variables using the quotient rule.

Key Terms and Formulas:

• Quotient Rule: $a^m \div a^n = a^{m-n}$

• Apply the rule to each variable and the coefficients.

Step-by-Step Guidance

1. Divide the coefficients: $2 \div 3$.

2. Apply the quotient rule to $x$: $x^3 \div x^2 = x^{3-2}$.

3. Apply the quotient rule to $y$: $y^2 \div y^5 = y^{2-5}$.

4. Combine the results to write the simplified expression.

Try solving on your own before revealing the answer!

Final Answer: $\frac{2x}{3y^3}$

We divided the coefficients and subtracted the exponents for each variable.

Q6. Simplify: $(5x^2y^3z^4)^2$

Background

Topic: Properties of Exponents

This question tests your ability to apply the power rule to a product of terms.

Key Terms and Formulas:

• Power Rule: $(a^m)^n = a^{mn}$

• Distribute the exponent to each factor inside the parentheses.

Step-by-Step Guidance

1. Apply the exponent to the coefficient: $(5)^2$.

2. Apply the power rule to each variable: $(x^2)^2 = x^{2 \cdot 2}$, $(y^3)^2 = y^{3 \cdot 2}$, $(z^4)^2 = z^{4 \cdot 2}$.

3. Combine all the results to write the expression in simplified form.

Try solving on your own before revealing the answer!

Final Answer: $25x^4y^6z^8$

We squared the coefficient and multiplied the exponents for each variable.