Spectrum Math Grade 8 Chapter 2 Lesson 4 Answer Key Using Roots to Solve Equations

Students can use the Spectrum Math Grade 8 Answer Key Chapter 2 Lesson 2.4 Using Roots to Solve Equations as a quick guide to resolve any of their doubts.

Spectrum Math Grade 8 Chapter 2 Lesson 2.4 Using Roots to Solve Equations Answers Key

Equations with exponential variables can be solved using the inverse operation. In this case, using roots will help to solve the problem.

c² = 121        Step 1: Evaluate the problem to find out which root to use. In this case, the exponent is 2, so you would use the square root as the inverse operation.
    Step 2: Find the root of both sides of the equation.
x = 11        Step 3: Solve the problem.

Solve each problem by using roots. Show your work and write fractions in simplest form.

Question 1.
a.
x² = \(\frac{16}{169}\)
x = _________
Answer: x = \(\frac{4}{13}\)
x² = \(\frac{16}{169}\)
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
\(\sqrt{x² }\) = \(\sqrt{\frac{16}{169}}\)
By simplification,
x = \(\frac{4}{13}\)

b. 729 = x³
x = ______
Answer: x = 9
729 = x³
As the exponent is 3, so use the cube root as the inverse operation.
Use root on both sides
\(\sqrt[3]{729}\) = \(\sqrt[3]{x³}\)
By simplification,
9 = x

c. x² = \(\frac{64}{625}\)
x = _________
Answer: x = \(\frac{8}{25}\)
x² = \(\frac{64}{625}\)
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
\(\sqrt{x² }\) = \(\sqrt{\frac{64}{625}}\)
By simplification,
x = \(\frac{8}{25}\)

Question 2.
a. 25 = x²
x = ________
Answer: x = 5
25 = x²
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
\(\sqrt{25}\) = \(\sqrt{x² }\)
By simplification,
5 = x

b. x² = \(\frac{25}{64}\)
x = _________
Answer: x = \(\frac{5}{8}\)
x² = \(\frac{25}{64}\)
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
\(\sqrt{x² }\) = \(\sqrt{\frac{25}{64}}\)
By simplification,
x = \(\frac{5}{8}\)

c. x³ = 512
x = _____
Answer: x  = 8
As the exponent is 3, so use the cube root as the inverse operation.
Use root on both sides
\(\sqrt[3]{x³}\) = \(\sqrt[3]{512}\)
By simplification,
x  = 8

Question 3.
a. \(\frac{9}{36}\) = x²
x = _________
Answer: x = \(\frac{3}{6}\)
\(\frac{9}{36}\) = x²
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
\(\sqrt{\frac{9}{36}}\) = \(\sqrt{x² }\)
By simplification,
\(\frac{3}{6}\) = x

b. x³ = 512
x = _________
Answer: x  = 8
x³ = 512
As the exponent is 3, so use the cube root as the inverse operation.
Use root on both sides
\(\sqrt[3]{x³}\) = \(\sqrt[3]{512}\)
By simplification,
x  = 8

c. x² + 2 = 38
x = ______
Answer: x = 6
x² + 2 = 38
x² = 38 – 2
x² = 36
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
\(\sqrt{x² }\) = \(\sqrt{36}\)
By simplification,
x = 6

Question 4.
a. 68 – 4 = x³
x = _____
Answer: x  = 4
68 – 4 = x³
64 = x³
As the exponent is 3, so use the cube root as the inverse operation.
Use root on both sides
\(\sqrt[3]{64}\) = \(\sqrt[3]{x³}\)
By simplification,
x  = 4

b. x² – 5 = 44
x = _____
Answer: x = 6.2449
x² – 5 = 44
x² = 44 – 5
x² = 39
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
\(\sqrt{x² }\) = \(\sqrt{39}\)
By simplification,
x = 6.2449

c. x³ + 4 = 5
x = _____
Answer: x  = 1
x³ + 4 = 5
x³ = 5 – 4
x³ = 1
As the exponent is 3, so use the cube root as the inverse operation.
Use root on both sides
\(\sqrt[3]{x³}\) = \(\sqrt[3]{1}\)
By simplification,
x  = 1

Equations with exponential variables can be solved using the inverse operation. In this case, using exponents will help to solve the problem.
= 6             Step 1: Evaluate the problem to decide which exponent to use. In this case, since we are solving for the square root, the appropriate exponent to use will be 2 (or square).
= 6²      Step 2: Square both sides of the equation.
x = 36                Step 3: Solve the problem.

Solve each problem by using roots. Show your work and write fractions in simplest form.

Question 1.
a. \(\sqrt{x}\)= 25
x = ______
Answer: x = 625
\(\sqrt{x}\)= 25
As the exponent is 2, so use the square root as the inverse operation.
Square both sides of the equation.
{\(\sqrt{x}\)}² = {25}²
By simplification,
x = 625

b. x = \(\sqrt{x}[latex]
x = _________
Answer: x = [latex]\sqrt{x}\)
x = \(\sqrt{x}\)

c.
= 6
x = _____
Answer: x = 216
\(\sqrt[3]{x}\) = 6
As the exponent is 3, so use the cube root as the inverse operation.
Square both sides of the equation.
{\(\sqrt[3]{x}\)}³ = {6}³
By simplification,
x = 216

Question 2.
a.
= 4
x = _____
Answer: x = 20
\(\sqrt{x-4}\)= 4
As the exponent is 2, so use the square root as the inverse operation.
Square both sides of the equation.
{\(\sqrt{x-4}\)}² = {4}²
By simplification,
x – 4= 16
x = 16 + 4
x = 20

b.
= 19
x = _____
Answer: x = 6859
\(\sqrt[3]{x}\) = 19
As the exponent is 3, so use the cube root as the inverse operation.
Square both sides of the equation.
{\(\sqrt[3]{x}\)}³ = {19}³
By simplification,
x = 6859

c. 7 = \(\sqrt{x}\)
x = _____
Answer: x = 49
7 = \(\sqrt{x}\)
\(\sqrt{x}\)= 7
As the exponent is 2, so use the square root as the inverse operation.
Square both sides of the equation.
{\(\sqrt{x}\)}² = {7}²
By simplification,
x = 49

Question 3.
a.
= 4
x = _____
Answer: x = 14
\(\sqrt[3]{78-x}\) = 4
As the exponent is 3, so use the cube root as the inverse operation.
Square both sides of the equation.
{\(\sqrt[3]{78-x}\)}³ = {4}³
By simplification,
78-x = 64
x  = 78 – 64
x = 14

b. 18 = \(\sqrt{x}\)
x = ____
Answer: x = 324
18 = \(\sqrt{x}\)
\(\sqrt{x}\)=18
As the exponent is 2, so use the square root as the inverse operation.
Square both sides of the equation.
{\(\sqrt{x}\)}² = {18}²
By simplification,
x = 324

c. 6 = \(\sqrt{42-x}\)
x = ______
Answer: x = 6
6 = \(\sqrt{42-x}\)
\(\sqrt{42-x}\)=6
As the exponent is 2, so use the square root as the inverse operation.
Square both sides of the equation.
{\(\sqrt{42-x}\)}² = {6}²
By simplification,
42-x = 36
x = 42-36
x = 6

Question 4.
a. 8 =
x = ______
Answer: x = 518
8 = \(\sqrt[3]{x-6}\)
\(\sqrt[3]{x-6}\) =8
As the exponent is 3, so use the cube root as the inverse operation.
Square both sides of the equation.
{\(\sqrt[3]{x-6}\)}³ = {8}³
By simplification,
x-6 = 512
x  = 512 + 6
x = 518

b. \(\sqrt{x}\) = 14
x = _____
Answer: x = 196
\(\sqrt{x}\)=14
As the exponent is 2, so use the square root as the inverse operation.
Square both sides of the equation.
{\(\sqrt{x}\)}² = {14}²
By simplification,
x = 196

c. 7 =
x = _____
Answer: x = 343
7 = \(\sqrt[3]{x}\)
\(\sqrt[3]{x}\) = 7
As the exponent is 3, so use the cube root as the inverse operation.
Square both sides of the equation.
{\(\sqrt[3]{x}\)}³ = {7}³
By simplification,
x = 343