Math  /  Numbers & Operations

QuestionM=[(23)6×(23)4]÷(23)8M=\left[\left(\frac{-2}{3}\right)^{6} \times\left(\frac{2}{-3}\right)^{4}\right] \div\left(\frac{2}{3}\right)^{8}
1. Verify that M=49\mathrm{M}=\frac{4}{9}.

Studdy Solution

STEP 1

1. The expression involves powers of fractions and division.
2. We need to simplify the expression step by step to verify if M=49 M = \frac{4}{9} .

STEP 2

1. Simplify each component of the expression.
2. Combine the components.
3. Simplify the entire expression.
4. Verify the result.

STEP 3

Simplify (23)6\left(\frac{-2}{3}\right)^{6}.
(23)6=(2)636=64729\left(\frac{-2}{3}\right)^{6} = \frac{(-2)^6}{3^6} = \frac{64}{729}

STEP 4

Simplify (23)4\left(\frac{2}{-3}\right)^{4}.
(23)4=24(3)4=1681\left(\frac{2}{-3}\right)^{4} = \frac{2^4}{(-3)^4} = \frac{16}{81}

STEP 5

Multiply the results from STEP_1 and STEP_2.
(64729)×(1681)=64×16729×81\left(\frac{64}{729}\right) \times \left(\frac{16}{81}\right) = \frac{64 \times 16}{729 \times 81}
Calculate the numerator and denominator:
64×16=102464 \times 16 = 1024 729×81=59049729 \times 81 = 59049
So, the product is:
102459049\frac{1024}{59049}

STEP 6

Simplify (23)8\left(\frac{2}{3}\right)^{8}.
(23)8=2838=2566561\left(\frac{2}{3}\right)^{8} = \frac{2^8}{3^8} = \frac{256}{6561}

STEP 7

Divide the result from STEP_3 by the result from STEP_4.
102459049÷2566561=102459049×6561256\frac{1024}{59049} \div \frac{256}{6561} = \frac{1024}{59049} \times \frac{6561}{256}
Calculate the division:
=1024×656159049×256= \frac{1024 \times 6561}{59049 \times 256}
Simplify the fraction:
=49= \frac{4}{9}

STEP 8

Verify the result.
The simplified expression equals:
49\frac{4}{9}
Thus, M=49 M = \frac{4}{9} is verified.

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