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Writing Radicals in Rational Form
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DEFINITIONS EXPONENT BASE
Base: The term/variable of which is being raised upon Exponent: The term/variable is raised by a term. AKA Power EXPONENT BASE
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THE EXPONENT
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𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚 NTH ROOT RULE m is the power (exponent)
n is the root or index a is the base or once in radical form a is the radicand 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚
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RULES Another way of writing 𝟐𝟓 is 𝟐𝟓 𝟏 𝟐 .
𝟐𝟓 is written in radical expression form. 𝟐𝟓 𝟏 𝟐 is written in rational exponent form.
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Evaluate 43/2 in radical form and simplify.
EXAMPLE 1 Evaluate 43/2 in radical form and simplify. 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚
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EXAMPLE 2 Evaluate 𝟒 𝟏 𝟐 in radical form and simplify. 4
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YOUR TURN Evaluate −𝟐𝟕 𝟐 𝟑 in radical form and simplify. 3 −27 2 =
3 − = (−3) 2 = 9
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𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚 EXAMPLE 3 Evaluate −𝟐𝟕 𝟒 𝟑 in radical form
and simplify. 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚 Use calculator to check Hint: Remember, the negative is OUTSIDE of the base
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EXAMPLE 4 Evaluate in radical form and simplify. or 𝑥−1 3 5 4𝑥 −1 3
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NTH ROOT RULE m is the power (exponent) n is the root a is the base
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EXAMPLE 5 Evaluate (27)–2/3 in radical form and simplify.
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EXAMPLE 6 Evaluate −𝟔𝟒 − 𝟐 𝟑 in radical form and simplify. 3 −64 −2 = −4 −2 = 1 − = 1 16
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YOUR TURN Evaluate 𝟐𝟓 𝟑𝟔 − 𝟏 𝟐 in radical form and simplify. 25 − −1 = = 6 5
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PROPERTIES OF EXPONENTS
Product of a Power: Power of a Power: Power of a Product: Negative Power Property: Quotient Power Property:
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If the BASES are the same, ADD the powers
EXAMPLE 7 Simplify Saying goes: BASE, BASE, ADD If the BASES are the same, ADD the powers
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EXAMPLE 8 Simplify 𝑥 • 𝑥 = 𝑥 = 𝑥 = 𝑥 5 6
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YOUR TURN Simplify 𝑥 • 𝑥 = 𝑥 = 𝑥 = 𝑥
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EXAMPLE 9 Simplify Saying goes: POWER, POWER, MULTIPLY
If the POWERS are near each other, MULTIPLY the powers – usually deals with PARENTHESES
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EXAMPLE 10 Simplify 2 𝑥 = 𝑥
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YOUR TURN 3 𝑥 1 4 𝑦 − 2 3 4 = 3 4 𝑥 1 4 4 𝑦 −2 3 4 = 81𝑥 𝑦 −8 3 =
Simplify 3 𝑥 𝑦 − = 𝑥 𝑦 − = 81𝑥 𝑦 − = 81𝑥 𝑦 8 3
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EXAMPLE 11 = 7 − 3 3 = 7 −1 = Simplify
Saying goes: When dividing an expression with a power, SUBTRACT the powers. = 7 − 3 3 = 7 −1 =
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EXAMPLE 12 Simplify 𝑥 𝑥 = 𝑥 − = 𝑥 − = 𝑥 1 6
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EXAMPLE 13 Simplify 𝑥 • 𝑥 1 4 𝑥 𝑥 1 5 𝑥 𝑥 𝑥 − 1 5 𝑥 7 12 𝑥 − 3 15 𝑥 •𝑥 7 12 𝑥 2 15 𝑥 = 𝑥 = 𝑥
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