Rules of Roots Combined Practice Problems & Exercises

Master combining powers and roots with step-by-step practice problems. Learn the 5 essential properties of radicals through interactive exercises and solutions.

📚Practice Combining Powers and Roots

Apply the five fundamental properties of roots and radicals
Convert between radical notation and exponential form using fractional exponents
Simplify products and quotients of square roots and higher-order radicals
Solve nested root expressions using the root of a root property
Combine multiple operations involving powers, roots, and basic arithmetic
Master the order of operations when working with radicals and exponents

Understanding Rules of Roots Combined

Complete explanation with examples

Understanding the combination of powers and roots is important and necessary.

First property:
$\sqrt a=a^{ 1 \over 2}$
Second property:
$\sqrt[n]{a^m}=a^{\frac{m}{n}}$
Third property:
$\sqrt{(a\times b)}=\sqrt{a}\times \sqrt{b}$

Fourth property:
$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Fifth property:
$\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\times m]{a}$

Detailed explanation

Practice Rules of Roots Combined

Test your knowledge with 68 quizzes

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

Examples with solutions for Rules of Roots Combined

Step-by-step solutions included

Exercise #1

$\frac{2^4}{2^3}=$

Step-by-Step Solution

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$

We apply it in the problem:

$\frac{2^4}{2^3}=2^{4-3}=2^1$

Remember that any number raised to the 1st power is equal to the number itself, meaning that:

$b^1=b$

Therefore, in the problem we obtain:

$2^1=2$

Therefore, the correct answer is option a.

Answer:

$2$

Video Solution

Exercise #2

$\frac{9^9}{9^3}=$

Step-by-Step Solution

Note that in the fraction and its denominator, there are terms with the same base, so we will use the law of exponents for division between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$

Let's apply it to the problem:

$\frac{9^9}{9^3}=9^{9-3}=9^6$

Therefore, the correct answer is b.

Answer:

$9^6$

Video Solution

Exercise #3

$(3^5)^4=$

Step-by-Step Solution

To solve the exercise we use the power property: $(a^n)^m=a^{n\cdot m}$

We use the property with our exercise and solve:

$(3^5)^4=3^{5\times4}=3^{20}$

Answer:

$3^{20}$

Video Solution

Exercise #4

$(6^2)^{13}=$

Step-by-Step Solution

We use the formula:

$(a^n)^m=a^{n\times m}$

Therefore, we obtain:

$6^{2\times13}=6^{26}$

Answer:

$6^{26}$

Video Solution

Exercise #5

Solve the exercise:

$(a^5)^7=$

Step-by-Step Solution

We use the formula:

$(a^m)^n=a^{m\times n}$

and therefore we obtain:

$(a^5)^7=a^{5\times7}=a^{35}$

Answer:

$a^{35}$

Video Solution

Frequently Asked Questions

Everything you need to know about Rules of Roots Combined

What are the 5 basic rules for combining roots and powers?

The five essential rules are: 1) √a = a^(1/2), 2) ∜[n]{a^m} = a^(m/n), 3) √(a×b) = √a × √b, 4) √(a/b) = √a / √b, and 5) ∜[n]{∜[m]{a}} = ∜[n×m]{a}. These properties allow you to convert between radical and exponential notation and simplify complex expressions.

How do you convert square roots to exponential form?

To convert square roots to exponential form, use the rule √a = a^(1/2). For higher-order roots, use ∜[n]{a^m} = a^(m/n) where n is the index and m is the power of the radicand.

When can you multiply square roots together?

You can multiply square roots with the same index using the property √a × √b = √(a×b). This works for any order of roots as long as they have the same index, such as ∜[3]{x} × ∜[3]{y} = ∜[3]{xy}.

What is the correct order of operations with roots and powers?

Follow this hierarchy: 1) Grouping symbols (parentheses, brackets), 2) Powers and roots (solved left to right when at same level), 3) Multiplication and division, 4) Addition and subtraction. Always solve operations inside radicals first before taking the root.

How do you simplify nested radicals like √(√a)?

Use the root of a root property: ∜[n]{∜[m]{a}} = ∜[n×m]{a}. For example, √(∜[3]{8}) = ∜[2×3]{8} = ∜[6]{8}. Multiply the indices together to create a single radical expression.

Can you divide square roots the same way you multiply them?

Yes, you can divide roots with the same index using √(a/b) = √a / √b. This property works in both directions, so you can either separate a quotient under one radical or combine separate radicals into one quotient.

What common mistakes should I avoid when combining roots and powers?

Common mistakes include: forgetting that √(a+b) ≠ √a + √b, mixing up the order of operations, incorrectly applying properties to roots with different indices, and not simplifying expressions completely. Always check if roots can be simplified before applying other operations.

How do you solve expressions with both positive and negative roots?

When dealing with even roots (like square roots), remember that √a² = |a| (absolute value). For odd roots like cube roots, ∜[3]{a³} = a without absolute value signs. Always consider the domain restrictions when working with real numbers.

Continue Your Math Journey

Master Rules of Roots Combined first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Square Root Rules Combining root laws

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

Applying the formula Calculating powers with negative exponents Complete the equation converting Negative Exponents to Positive Exponents Factoring Out the Greatest Common Factor (GCF) Factorization Identify the greater value More than one unknown Multiplying Exponents with the same base Number of terms A power law Presenting powers in the denominator as powers with negative exponents Presenting powers with negative exponents as fractions Presenting powers with negative exponents as fractions Monomial Single Variable Trinomial Binomial Two Variables Using laws of exponents with parameters Using multiple rules Using the laws of exponents Using variables Variable in the base of the power Variable in the exponent of the power Variables in the exponent of the power Worded problems The power of zero Applying the formula Identify the greater value Number of terms Same base and different indicator Solving the equation Using multiple rules Using variables

Rules of Roots Combined Practice Problems & Exercises

Understanding Rules of Roots Combined

Practice Rules of Roots Combined

Examples with solutions for Rules of Roots Combined

Frequently Asked Questions

What are the 5 basic rules for combining roots and powers?

How do you convert square roots to exponential form?

When can you multiply square roots together?

What is the correct order of operations with roots and powers?

How do you simplify nested radicals like √(√a)?

Can you divide square roots the same way you multiply them?

What common mistakes should I avoid when combining roots and powers?

How do you solve expressions with both positive and negative roots?

More Rules of Roots Combined Questions

Applying Combined Exponents Rules