Video: Pack 5 • Paper 2 • Question 16

Pack 5 • Paper 2 • Question 16

04:06

Video Transcript

Work out the exact value of 𝑥 from the following equation: 27 to the power of 𝑥 divided by nine to the power of three-sevenths is equal to three to the power of three-quarters.

In order to solve this problem, we need to use our laws of indices. Firstly, 𝑎 to the power of 𝑥 multiplied by 𝑎 to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 plus 𝑦. When we are multiplying, we can add our powers. 𝑎 to the power of 𝑥 divided by 𝑎 to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 minus 𝑦. When we are dividing, we can subtract the powers. And finally, 𝑎 to the power of 𝑥 to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 multiplied by 𝑦. In this case, we multiply the powers or indices.

In order for the laws of indices to work, our base number or letter — in this case 𝑎 — must be the same. 27 is equal to three cubed and nine is equal to three squared. This means that the equation can be rewritten three cubed to the power of 𝑥 divided by three squared to the power of three-sevenths is equal to three to the power of three-quarters.

Using one of the laws of indices, we can now multiply the powers on the first two terms. Three multiplied by 𝑥 is three 𝑥 and two multiplied by three-sevenths is six-sevenths. We are left with three to the power of three 𝑥 divided by three to the power of six-sevenths is equal to three to the power of three-quarters. Multiplying both sides of this equation by three to the power of six-sevenths gives us three to the power of three 𝑥 is equal to three to the power of three-quarters multiplied by three to the power of six-sevenths.

Using one of the laws of indices again, as we are multiplying three to the power of three-quarters by three to the power of six-sevenths, we can add the powers. This gives us three to the power of three-quarters plus six-sevenths. As the base numbers at both sides of this equation are the same, the powers must be equal: three 𝑥 must be equal to three-quarters plus six-sevenths.

In order to add two fractions, we need to find the common denominator — in this case 28 as the lowest number in the four and seven times table is 28. We’ve multiplied the first fraction by seven. And whatever we do to the bottom or denominator, we must do it to the top, the numerator. Three multiplied by seven is 21. Likewise, in the second fraction, six multiplied by four is equal to 24. This means that three-quarters plus six-sevenths is equal to forty-five twenty-eighths or 45 over 28.

If three 𝑥 is equal to 45 over 28, we need to divide both sides of the equation by three to calculate 𝑥. Dividing both sides by three gives us 𝑥 is equal to 15 over 28 or fifteen twenty-eighths.

This means that the exact value of 𝑥 that satisfies the equation 27 to the power of 𝑥 divided by nine to the power of three-sevenths is equal to three to the power of three-quarters is fifteen twenty-eighths or 15 over 28.

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