Question Video: Evaluating an Expression Involving Powers of Rational Numbers | Nagwa Question Video: Evaluating an Expression Involving Powers of Rational Numbers | Nagwa

Question Video: Evaluating an Expression Involving Powers of Rational Numbers Mathematics • First Year of Preparatory School

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Evaluate the expression ((2/3)³ × (6/5)²) ÷ (4/5)⁴, giving your answer as a fraction in its simplest form.

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Video Transcript

Evaluate the expression two-thirds cubed times six-fifths squared all divided by four-fifths raised to the fourth power, giving your answer as a fraction in its simplest form.

In this question, we are given an expression involving the product and quotient of rational numbers raised to positive integer exponents. And we need to evaluate the expression, giving our answer as a fraction in its simplest form. To do this, let’s start by evaluating the exponents. In this case, we are raising fractions to positive integer exponents. There are a few ways of evaluating these exponents. The easiest way is to recall that 𝑎 over 𝑏 all raised to the power of 𝑛 is equal to 𝑎 raised to the power of 𝑛 over 𝑏 raised to the power of 𝑛, provided that these are all well defined.

We can apply this result to evaluate each of the fractions raised to an exponent to get two cubed over three cubed times six squared over five squared all divided by four raised to the fourth power over five raised to the fourth power. We can then recall that we multiply fractions by multiplying their numerators and denominators separately. This allows us to rewrite the expression as shown.

We could simplify at this point. However, we can also note that our expression involves division by a fraction. We know that dividing by a fraction is the same as multiplying by its reciprocal. So, instead of dividing by four raised to the fourth power over five raised to the fourth power, we can multiply by its reciprocal to obtain the following expression. We can then once again multiply the numerators and denominators separately. This gives us two cubed times six squared times five raised to the fourth power all divided by three cubed times five squared times four raised to the fourth power.

To help us simplify this expression, we can rewrite each factor to be in terms of powers of primes. We can calculate that six squared is equal to three squared times two squared and four raised to the fourth power is equal to two raised to the eighth power. This allows us to rewrite the expression as shown.

We are now ready to cancel shared factors in the numerator and denominator. We can do this directly, or we can use the quotient rule for exponents, which tells us that 𝑎 raised to the power of 𝑚 over 𝑎 raised to the power of 𝑛 is equal to 𝑎 raised to the power of 𝑚 minus 𝑛. In other words, when taking the quotient of exponential expressions with the same base, we can raise the base to the difference in the exponents.

We can cancel five shared factors of two in the numerator and denominator to be left with a factor of two cubed in the denominator. Similarly, we can cancel a shared factor of three squared in the numerator and denominator to be left with a factor of three in the denominator. Finally, we can cancel a shared factor of five squared in the numerator and denominator to be left with a factor of five squared in the numerator. We can then evaluate the numerator and denominator to get 25 over 24. We cannot simplify the fraction any further, so this is our final answer.

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