Exercise practice 1: Power Play

3.1 Exercise practice 1: Figure it Out, page 4

This page covers the first Figure it Out section, on exponential form and prime-factor powers.

3.1.1 E1.1 Exponential form

Question. Write each product in exponential form: (i) 6 x 6 x 6 x 6; (ii) y x y; (iii) b x b x b x b; (iv) 5 x 5 x 7 x 7 x 7; (v) 2 x 2 x a x a; (vi) a x a x a x c x c x c x c x d.

Answer. (i) 6^4; (ii) y^2; (iii) b^4; (iv) 5^2 x 7^3; (v) 2^2 x a^2; (vi) a^3 x c^4 x d.

Key steps. Count how many times each base appears. In (iv), 5 appears twice and 7 appears three times. In (vi), a appears three times, c four times and d only once, so d keeps no exponent. When the same base appears once, the exponent 1 is not written.

3.1.2 E1.2 Prime factors with exponents

Question. Write 648, 405, 540 and 3600 as products of powers of their prime factors.

Answer. 648 gives 2^3 x 3^4; 405 gives 3^4 x 5; 540 gives 2^2 x 3^3 x 5; and 3600 gives 2^4 x 3^2 x 5^2.

Key steps. Divide repeatedly by the smallest prime that divides the number. For 648: 648 = 8 x 81 = 2^3 x 3^4. For 405: 405 = 5 x 81 = 3^4 x 5. For 540: 540 = 4 x 135 = 2^2 x 3^3 x 5. For 3600: 3600 = 36 x 100 = 2^2 x 3^2 x 2^2 x 5^2 = 2^4 x 3^2 x 5^2. Check each product back to the original number.

3.1.3 E1.3 Numerical values

Question. Find the value of (i) 2 x 10^3; (ii) 7^2 x 2^3; (iii) 3 x 4^4; (iv) (-3)^2 x (-5)^2; (v) 3^2 x 10^4; (vi) (-2)^5 x (-10)^6.

Answer. The first value is 2000, the second 392, the third 768, the fourth 225, the fifth 90000, and the sixth -32000000.

Key steps. Evaluate each power first, then multiply. (ii) 49 x 8 = 392. (iii) 3 x 256 = 768. (iv) 9 x 25 = 225; the two negative bases have even exponents, so the product is positive. (v) 9 x 10000 = 90000. (vi) (-2)^5 = -32 and (-10)^6 = 1000000; the product is negative, -32000000.

3.2 Method recap for exponential form

Three habits prevent most errors in this section. First, identify the base before counting. In 5 x 5 x 7 x 7 x 7 the bases 5 and 7 are different, so they cannot be merged into one power. Second, count each repeat separately and write one exponent per base. Third, check by expanding the answer mentally: 5^2 x 7^3 means two 5s and three 7s, which matches the original product.

For prime factorisation, keep dividing by the smallest prime that goes in evenly. Start with 2, then 3, then 5, and continue until the quotient is 1. The list of divisors gives the prime factors, and grouping equal divisors gives the exponents. For example, dividing 540 by 2 gives 270, by 2 again gives 135, by 3 gives 45, by 3 gives 15, by 3 gives 5, so 540 = 2^2 x 3^3 x 5.

When evaluating expressions, the order of operations matters. A product such as 3 x 4^4 means three times the fourth power of 4, not the fourth power of 3 x 4. Work out the power first, then multiply by the coefficient. For negative bases, check the exponent's parity before multiplying.

3.3 Quick self-check

Try these checks after finishing the exercise.

• Does 6 x 6 x 6 x 6 equal 6^4 or 4^6? It equals 6^4, because 6 is repeated four times.
• Is 5^2 x 7^3 equal to 35^5? No, because the bases are different; 35^5 would mean five factors of 35.
• Is 3600 = 2^4 x 3^2 x 5^2? Expand: 16 x 9 x 25 = 3600, so yes.
• Is (-3)^2 x (-5)^2 positive? Yes, each square is positive, so the product is positive.
• Is 2 x 10^3 equal to 2000 or 8000? It is 2000; 8000 would be 2^3 x 10^3.

3.4 Checking the numbers

A calculator is useful for the final arithmetic, but it cannot choose the method. Estimate the answer before pressing the buttons. For 3 x 4^4, note that 4^4 = 256 is a little above 250, so three times it is a little above 750; the exact answer 768 fits this estimate. For (-2)^5, the negative sign survives because the exponent is odd, so the answer must be negative.

When several powers are multiplied, group the same bases first. The expression 2^4 x 3^6 x 2^3 becomes 2^7 x 3^6, and only then is evaluated. Grouping keeps the numbers small and makes a slip easier to see. If a question asks for exponential form, do not expand the powers; leave them with their exponents.

A final check is to compare the exponent counts with the size of the answer. Numbers with larger exponents should be larger when the bases are the same. This is the same reasoning used to compare 2^100 with 100^2: the repeated doubling wins long before the hundredth step.