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Math  /  Numbers & Operations

Question1) (45)2\left(\frac{4}{5}\right)^{2}; 2) (−23)2\left(-\frac{2}{3}\right)^{2}; 3) (112)2\left(1 \frac{1}{2}\right)^{2}; 4) (−214)2\left(-2 \frac{1}{4}\right)^{2};

Studdy Solution

STEP 1

1. Each expression involves raising a rational or mixed number to the power of 2.
2. Simplification of these expressions requires basic algebraic manipulation and conversion between mixed numbers and improper fractions where necessary.
3. The final expressions should be in the simplest rational form.

STEP 2

1. Simplify (45)2\left(\frac{4}{5}\right)^{2}.
2. Simplify (−23)2\left(-\frac{2}{3}\right)^{2}.
3. Simplify (112)2\left(1 \frac{1}{2}\right)^{2}.
4. Simplify (−214)2\left(-2 \frac{1}{4}\right)^{2}.

STEP 3

Calculate (45)2\left(\frac{4}{5}\right)^{2}.
(45)2=4252=1625 \left(\frac{4}{5}\right)^{2} = \frac{4^2}{5^2} = \frac{16}{25}

STEP 4

Calculate (−23)2\left(-\frac{2}{3}\right)^{2}.
(−23)2=(−2)232=49 \left(-\frac{2}{3}\right)^{2} = \frac{(-2)^2}{3^2} = \frac{4}{9}

STEP 5

Convert the mixed number 1121 \frac{1}{2} to an improper fraction.
112=32 1 \frac{1}{2} = \frac{3}{2}

STEP 6

Calculate (32)2\left(\frac{3}{2}\right)^{2}.
(32)2=3222=94 \left(\frac{3}{2}\right)^{2} = \frac{3^2}{2^2} = \frac{9}{4}

STEP 7

Convert the mixed number −214-2 \frac{1}{4} to an improper fraction.
−214=−94 -2 \frac{1}{4} = -\frac{9}{4}

STEP 8

Calculate (−94)2\left(-\frac{9}{4}\right)^{2}.
(−94)2=(−9)242=8116 \left(-\frac{9}{4}\right)^{2} = \frac{(-9)^2}{4^2} = \frac{81}{16}
Solution:
1. (45)2=1625\left(\frac{4}{5}\right)^{2} = \frac{16}{25}
2. (−23)2=49\left(-\frac{2}{3}\right)^{2} = \frac{4}{9}
3. (112)2=94\left(1 \frac{1}{2}\right)^{2} = \frac{9}{4}
4. (−214)2=8116\left(-2 \frac{1}{4}\right)^{2} = \frac{81}{16}

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