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Negative Exponents, Reciprocals, and The Exponent Laws

Negative Exponents, Reciprocals, and The Exponent Laws. Relating Negative Exponents to Reciprocals, and Using the Exponent Laws. Today’s Objectives. Students will be able to demonstrate an understanding of powers with integral and rational exponents, including:

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Negative Exponents, Reciprocals, and The Exponent Laws

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  1. Negative Exponents, Reciprocals, and The Exponent Laws Relating Negative Exponents to Reciprocals, and Using the Exponent Laws

  2. Today’s Objectives • Students will be able to demonstrate an understanding of powers with integral and rational exponents, including: • Explain, using patterns, why x-n= 1/xn, x ≠ 0 • Apply the exponent laws • Identify and correct errors in a simplification of an expression that involves powers

  3. Reciprocals • Any two numbers that have a product of 1 are called reciprocals • 4 x ¼ = 1 • 2/3 x 3/2 = 1 • Using the exponent law: am x an = am+n, we can see that this rule also applies to powers • 5-2 x 52 = 5-2+2 = 50 = 1 • Since the product of these two powers is 1, 5-2 and 52 are reciprocals • So, 5-2 = 1/52, and 1/5-2 = 52 • 5-2 = 1/25

  4. Powers with Negative Exponents • When x is any non-zero number and n is a rational number, x-n is the reciprocal of xn • That is, x-n = 1/xn and 1/x-n = xn, x ≠ 0 • This is one of the exponent laws:

  5. Example 1: Evaluating Powers with Negative Integer Exponents • Evaluate each power: • 3-2 • 3-2 = 1/32 • 1/9 • (-3/4)-3 • (-3/4)-3 = (-4/3)3 • -64/27 • We can apply this law to evaluate powers with negative rational exponents as well • Look at this example: • 8-2/3 • The negative sign represents the reciprocal, the 2 represents the power, and the 3 represents the root

  6. Example 2: Evaluating Powers with Negative Rational Exponents • Remember from last class that we can write a rational exponent as a product of two or more numbers • The exponent -2/3 can be written as (-1)(1/3)(2) • Evaluate the power: • 8-2/3 • 8-2/3 = 1/82/3 = 1/(3√8)2 • 1/22 • 1/4 • Your turn: Evaluate (9/16)-3/2 • (16/9)3/2 = (√16/9)3 = (4/3)3 = 64/27

  7. Exponent Laws • Product of Powers • am x an = am+n • Quotient of Powers • am/an = am-n, a ≠ 0 • Power of a Power • (am)n = amn • Power of a Product • (ab)m = ambm • Power of a Quotient • (a/b)m = am/bm, b ≠ 0

  8. Applying the Exponent Laws • We can use the exponent laws to simplify expressions that contain rational number bases • When writing a simplified power, you should always right your final answer with a positive exponent • Example 3: Simplifying Numerical Expressions with Rational Number Bases • Simplify by writing as a single power: • [(-3/2)-4]2 x [(-3/2)2]3 • First, use the power of a power law: • For each power, multiply the exponents • (-3/2)(-4)(2) x (-3/2)(2)(3) = (-3/2)-8 x (-3/2)6

  9. Example 3 • Next, use the product of powers law • (-3/2)-8+6 = (-3/2)-2 • Finally, write with a positive exponent • (-3/2)-2 = (-2/3)2 • Your turn: Simplify (1.43)(1.44)/1.4-2 • 1.43+4/1.4-2 = 1.47/1.4-2 = 1.47-(-2) = 1.49 • We will also be simplifying algebraic expressions with integer and rational exponents

  10. Example 4 • Simplify the expression 4a-2b2/3/2a2b1/3 • First use the quotient of powers law • 4/2 x a-2/a2 x b2/3/b1/3 = 2 x a(-2)-2 x b2/3-1/3 • 2a-4b1/3 • Then write with a positive exponent • 2b1/3/a4 • Your turn: • Simplify (100a/25a5b-1/2)1/2 • (100/25 x a1/a5 x 1/b-1/2)1/2 • (4a1-5b1/2)1/2= (4a-4b1/2)1/2 • 41/2a(-4)(1/2)b(1/2)(1/2) = 2a-2b1/4 • 2b1/4/a2

  11. Review

  12. Roots and Powers Assignment • Complete the Assignment • Due: Friday, beginning of class • Extra Practice: • Chapter Review, pg. 246 – 249 • Review: • Chapter 1-4, pg. 252 - 253

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