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Even Powers

Part of the Radicals & Rational Exponents section of Coddy's Algebra 2 journey. Lesson 28 of 40.

An even power hides the sign of its base: ${2}^{4}$ and ${\left(-2\right)}^{4}$ are both 16. So ${x}^{4} = 16$ has two real answers, x = 2 and x = -2. The fourth root of 16 is 2, and the negative answer comes back from the sign the power hid.

This is the same idea as ${x}^{2} = 49$ giving 7 and -7, carried to higher even powers. Forgetting the negative answer is the usual mistake, so always ask first whether the power is even. If the power is not alone, get it alone before taking the root: $3{x}^{4} = 48$ becomes ${x}^{4} = 16$ .

An even power of a real number is never negative, so ${x}^{4} = -16$ has no real answer at all.

Challenge

Solve $2{x}^{4}-162 = 0$ . Get the power alone, then find every real answer.

Try it yourself

2x^{4}-162 = 0

Get $x$ alone

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Radicals & Rational Exponents

1Simplest Form

Perfect Square Factors Like Radicals Simplify First Recap: Adding Radicals

4Rationalizing

One Root Below A Number and a Root The Conjugate Below Recap: Rationalizing

7Powers and Roots Equations

Odd PowersEven PowersFractional Powers Two Thirds Power Recap: Power Equations

2Letters and Cube Roots

Letters Under the Root Cube Roots Recap: Simplifying Radicals

5Rational Exponents

One Over n m Over n Negative Exponents Exponent Rules Still Work Recap: Rational Exponents

8Radical Functions

Domain of a Root Evaluating Inverse of a Square Recap: Radical Functions

3Multiplying Radicals

Root Times Root Distributing a Root Two Brackets Conjugate Pairs Recap: Multiplying Radicals

6Radical Equations

Square Both Sides Isolate the Root First A False Answer Cube Root Equations Recap: Radical Equations

9Final Challenges

Final: Simplify and Add Final: Clear the Root Final: The Exponent Final: Check It Final: The Falling Ball