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Simplify numerical expressions involving exponents and roots

Numerical simplification involves interpreting powers as repeated multiplication and roots as inverse operations, including principal square and cube roots of perfect powers, such as 49=7\sqrt{49}=7 and 273=3\sqrt[3]{27}=3. Expressions are evaluated using grouping symbols and the order of operations, with careful attention to signs and the distinction between (−3)2(-3)^2 and −32-3^2. The focus remains on numerical cases, excluding variable exponents, rational or negative exponents, and rationalizing general radicals.

Detailed Explanation: Simplify numerical expressions involving exponents and roots

To simplify numerical expressions with exponents and roots:

  1. Evaluate powers and roots.
  2. Perform multiplication or division.
  3. Add or subtract from left to right.
  4. Pay attention to parentheses and signs.

Worked example

Simplify:

(−3)2−249+273(-3)^2 - 2\sqrt{49}+\sqrt[3]{27}

Step 1: Evaluate the exponent and roots.

The parentheses mean the entire number −3-3 is squared:

(−3)2=9(-3)^2=9

The principal square root of 4949 is 77, and the cube root of 2727 is 33:

49=7and273=3\sqrt{49}=7 \qquad\text{and}\qquad \sqrt[3]{27}=3

Substitute these values:

9−2(7)+39-2(7)+3

Step 2: Multiply.

2(7)=142(7)=14

So the expression becomes:

9−14+39-14+3

Step 3: Add and subtract from left to right.

9−14=−59-14=-5 −5+3=−2-5+3=-2

Therefore,

−2\boxed{-2}

Remember the difference between (−3)2(-3)^2 and −32-3^2:

(−3)2=9but−32=−(32)=−9(-3)^2=9 \qquad\text{but}\qquad -3^2=-(3^2)=-9

Parentheses determine whether the negative sign is included in the exponent.

Learn by doing: Simplify numerical expressions involving exponents and roots

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    Radicals - Division with Mixed Index and Power of Radicand (Fraction) - Radical over Integer

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