6th Grade exponents and square roots worksheets: Free & Printable

Exponents and square roots are two ways of expressing repeated multiplication. For example, 2³ means 2 multiplied by itself 3 times, which is 8. Similarly, the square root of 16 means the number multiplied by itself gives 16, which is 4. Exponents and square roots are also called powers and radicals, respectively.

Exponents and square roots are important in 6th-grade math because they help students understand patterns, relationships, and properties of numbers. They also prepare them for more advanced topics in algebra, geometry, and science. For example, we can use exponents to represent very large or tiny numbers, such as the size of the universe or the mass of an atom.

Using the Pythagorean Theorem, we can also use square roots to find the length of the sides of a right triangle.

There are some rules that we must follow when working with exponents. These rules help us to simplify expressions and perform operations with exponents. Here are some of the most common rules of exponents:

• The product rule: Add the exponents when multiplying two powers with the same base. For example, 2³ x 2⁴ = 2⁽³⁺⁴) = 2⁷.
• The quotient rule: Subtract the exponents when dividing two powers with the same base. For example, 2⁵ / 2² = 2^(5-2) = 2³.
• The power rule: When raising a power to another power, multiply the exponents. For example, (2³)⁴ = 2^(3x4) = 2¹².
• The zero exponent rule: Any number raised to zero power equals one. For example, 2⁰ = 1.
• The negative exponent rule: Any number raised to a negative power equals the reciprocal of that number raised to the positive power. For example, 2^-3 = 1 / 2³ = 1/8.

To simplify expressions with exponents, we apply the rules of exponents and use the order of operations (PEMDAS). For example, to simplify the expression (2³ x 4-²)² / (8 x 2^-1), we can follow these steps:

1. Apply the power rule to the numerator and denominator: (2⁶ x 4^-4) / (8 x 2^-1)
2. Apply the product rule to the numerator and denominator: 2^(6-4) x 4^-4 / (8 x 2^-1)
3. Simplify the powers: 2² x (1/16) / (8 x (1/2))
4. Multiply and divide from left to right: (4 x (1/16)) / (8 x (1/2))
5. Cancel out common factors: (1/4) / (4/1)
6. Invert and multiply: (1/4) x (1/4)
7. Simplify the fraction: 1/16

Another important skill for 6th-grade math is finding the square root of a number. The square root of a number is the number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 x 5 = 25.

A perfect square is a number that has a whole number as its square root. For example, 36 is a perfect square because the square root of 36 is 6, which is a whole number. Some other examples of perfect squares are 1, 4, 9, 16, and 49.

To find the square root of a perfect square, you can use a calculator or try to find the factors of the number that are equal. For example, to find the square root of 64, you can look for two factors of 64 that are the same: 8 x 8 = 64. So, the square root of 64 is 8.

Sometimes, you may encounter equations that involve both exponents and square roots. For example:

x² = 49

To solve this equation, you need to find the value of x that makes the equation true. One way to do this is to take the square root of both sides of the equation. This will give you:

x = ±√49

The symbol ± means plus or minus. This means there are two possible x values: positive and negative. In this case, x can be either +7 or -7 because both numbers satisfy the equation.

x² = 49

(+7)² = 49

(-7)² = 49

To check our answer, we can plug the values of x into the original equation and see if they make it true.

When solving equations with exponents and square roots, we must be careful with the signs and the order of operations. Remember to follow these rules:

• When taking the square root of both sides of an equation, include both the positive and negative solutions.
• When raising a negative number to an even power, use parentheses. For example, (-3)² = 9, but -3² = -9.
• When raising a negative number to an odd power, we do not need parentheses. For example, (-3)³ = -27 and -3³ = -27.
• When simplifying expressions with exponents and square roots, follow the order of operations: parentheses, exponents, multiplication/division, addition/subtraction (PEMDAS).

To help your 6^th graders practice solving equations with exponents and square roots, we have prepared some more free and printable worksheets that you can download from Mathskills4kids.com and print at home or school. Each worksheet has an answer key for your convenience.

• Write multiplication expressions using exponents: This worksheet helps students practice writing repeated multiplication as powers. For example, 5 x 5 x 5 = 5³.
• Evaluate powers: This worksheet helps students practice finding the value of power by multiplying the base by itself as many times as the exponent. For example, 2⁴ = 2 x 2 x 2 x 2 = 16.
• Write powers of ten with exponents: This worksheet helps 6^th graders practice writing powers of ten using exponents. For example, 10,000 = 10⁴.
• Find the missing exponent or base: This worksheet helps students practice finding the missing exponent or base in a power equation. For example, 3^? = 27 or ?² = 49.
• Powers with decimal bases: This worksheet helps Grade 6 students practice finding the value of powers with decimal bases. For example, 0.1³ = 0.001.
• Powers with fractional bases: This worksheet helps children practice finding the value of powers with fractional bases. For example, (1/2)³ = 1/8.
• Square roots of perfect squares: This worksheet helps learners practice finding the square root of a perfect square. A perfect square is a number that can be written as the product of two equal factors. For example, 64 is a perfect square because it can be written as 8 x 8 or (8)². The square root of 64 is 8 because 8 x 8 = 64.
• Estimate square roots: This worksheet helps 6^th graders practice estimating the square root of a non-perfect square by finding the two closest perfect squares and using them as a range. For example, to estimate the square root of 50, we can find that 49 and 64 are the closest perfect squares to 50. The square root of 49 is 7, and the square root of 64 is 8, so the square root of 50 is between 7 and 8.