Video: Pack 3 β€’ Paper 2 β€’ Question 12

Pack 3 β€’ Paper 2 β€’ Question 12

03:29

Video Transcript

π‘₯ to the power of seven divided by π‘₯ to the power of 𝑦 is equal to π‘₯ to the power of 10. Find the value of 𝑦.

To answer this question, we need to recall what happens to the powers when we’re dividing two terms with the same base. We subtract the powers. π‘₯ to the power of π‘Ž divided by π‘₯ to the power of 𝑏 is equal to π‘₯ to the power of π‘Ž minus 𝑏. This means that the left-hand side of the equation can be written as π‘₯ to the power of seven minus 𝑦.

As the bases of the two sides of the equation are the same β€” they’re both π‘₯ β€” we can equate the powers. This gives seven minus 𝑦 is equal to 10, which is a linear equation that we can solve in order to find the value of 𝑦.

As the coefficient of 𝑦 is negative, we can begin by adding 𝑦 to both sides of the equation, giving seven is equal to 𝑦 plus 10. Next, we subtract 10 from both sides to solve for 𝑦, giving negative three is equal to 𝑦.

The second part of the question says three squared to the power of π‘˜ is equal to three to the power of 12. Find the value of π‘˜.

To answer this part of the question, we need to recall what happens when you raise a number or a letter to a power and then another power. We multiply the powers together. π‘₯ to the power of π‘Ž to the power of 𝑏 is π‘₯ to the power of π‘Žπ‘. This means that the left-hand side of the equation can be written as three to the power of two π‘˜. So we have three to the power of two π‘˜ is equal to three to the power of 12.

As the bases on both sides of the equation are the same, we can again equate the powers, giving two π‘˜ is equal to 12. Dividing both sides by two solves the equation and gives the value of π‘˜. π‘˜ is equal to six.

The third part of the question says eight to the power of 𝑝 multiplied by 16 to the power of π‘ž can be written in the form two to the power of π‘Ÿ. Show that π‘Ÿ is equal to three 𝑝 plus four π‘ž.

We’ve been asked to express this product in the form two to the power of π‘Ÿ, that is, as a power of two. To do so, we first recall that both eight and 16 are powers of two. Eight is equal to two cubed. And 16 is equal to two to the power of four. We can replace eight and 16 with two cubed and two to the power of four, giving two cubed to the power of 𝑝 multiplied by two to the power of four to the power of π‘ž.

If we recall the result that we used in the second part of this question, we know how to simplify the brackets. We can multiply together the powers inside and outside each bracket. This gives two to the power of three 𝑝 multiplied by two to the power of four π‘ž.

Finally, we need to recall what happens to the powers when we multiply together two terms with the same base. We add the powers. π‘₯ to the power of π‘Ž multiplied by π‘₯ to the power of 𝑏 is equal to π‘₯ to the power of π‘Ž plus 𝑏. Adding together the powers gives two to the power of three 𝑝 plus four π‘ž. And if we look back at the question, this is what we wanted to show. We’ve expressed the product in the form two to the power of π‘Ÿ, where π‘Ÿ is equal to three 𝑝 plus four π‘ž.

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