Math

QuestionCalculate (56)2÷(5849)\left(\frac{5}{6}\right)^{2} \div\left(\frac{5}{8}-\frac{4}{9}\right).

Studdy Solution

STEP 1

Assumptions1. We are given the expression (56)÷(5849)\left(\frac{5}{6}\right)^{} \div\left(\frac{5}{8}-\frac{4}{9}\right). We will follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is often abbreviated as PEMAS.

STEP 2

First, we need to simplify the expression inside the parentheses. This involves squaring the fraction 56\frac{5}{6}.
(56)2=(56)×(56)\left(\frac{5}{6}\right)^{2} = \left(\frac{5}{6}\right) \times \left(\frac{5}{6}\right)

STEP 3

Now, calculate the square of the fraction 56\frac{5}{6}.
(56)2=2536\left(\frac{5}{6}\right)^{2} = \frac{25}{36}

STEP 4

Next, we need to simplify the expression inside the parentheses in the denominator. This involves subtracting the fraction 49\frac{4}{9} from the fraction 8\frac{}{8}.
849\frac{}{8} - \frac{4}{9}

STEP 5

To subtract fractions, we need to have a common denominator. The least common denominator (LCD) of8 and9 is72.
58×9949×88=45723272\frac{5}{8} \times \frac{9}{9} - \frac{4}{9} \times \frac{8}{8} = \frac{45}{72} - \frac{32}{72}

STEP 6

Now, subtract the fractions.
45723272=1372\frac{45}{72} - \frac{32}{72} = \frac{13}{72}

STEP 7

Now that we have simplified both the numerator and the denominator, we can simplify the entire expression. This involves dividing the fraction 2536\frac{25}{36} by the fraction 1372\frac{13}{72}.
2536÷1372\frac{25}{36} \div \frac{13}{72}

STEP 8

Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
2536×7213\frac{25}{36} \times \frac{72}{13}

STEP 9

Now, multiply the fractions.
2536×7213=180468\frac{25}{36} \times \frac{72}{13} = \frac{180}{468}

STEP 10

implify the fraction 1800468\frac{1800}{468} to its lowest terms.
1800468=7513\frac{1800}{468} = \frac{75}{13}The simplified form of the given expression is 7513\frac{75}{13}.

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