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Which of the following is equal to two and three-quarters squared?
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Is it (A) seven and nine sixteenths, (B) four and nine sixteenths, (C) five and a half, (D) two and nine sixteenths, or (E) twenty-two eighths?
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Our first step is to turn the mixed number two and three-quarters into an improper or top-heavy fraction.
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Two is equal to eight-quarters as there are four quarters in one whole.
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This means that two and three quarters is the same as eight-quarters plus three-quarters.
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When adding two fractions with the same denominator, we just add the numerators.
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Therefore, eight quarters plus three quarters is equal to eleven-quarters.
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A quicker way to change a mixed number into an improper fraction is to multiply the whole number, in this case two, by the denominator four and then add the numerator.
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Two multiplied by four plus three is equal to 11.
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Therefore, two and three-quarters is equal to eleven-quarters.
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We can therefore write two and three-quarters squared as eleven-quarters squared.
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Any fraction 𝑎 over 𝑏 that is raised to the power of 𝑥 can be rewritten as 𝑎 to the power of 𝑥 divided by 𝑏 to the power of 𝑥.
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This means that 11 over four or eleven-quarters squared is the same as 11 squared divided by four squared.
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11 squared is equal to 121, and four squared is equal to 16.
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Therefore, 11 squared divided by four squared is equal to 121 over 16.
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We now need to turn this top-heavy or improper fraction back into a mixed number.
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In order to do this, our first step is to work out how many 16s go into 121.
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Six times 16 is equal to 96.
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Seven times 16 is equal to 112.
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And eight times 16 is equal to 128.
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We can therefore say that 16 divides into 121 seven times with a remainder of nine.
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As 121 minus 112 is equal to nine, the improper fraction 121 over 16 can be rewritten as seven and nine sixteenths.
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This means that the correct answer was A: seven and nine sixteenths is equal to two and three-quarters squared.