(a) Write \( \frac{1}{2 \times 2 \times 2 \times 2 \times 2} \) as a power of 2.
(b) (i) \( 3^{18} \div 3^t = 3^6 \) Find the value of t.
(ii) Simplify \( 8w^{10} \times 6w^5 \).
▶️ Answer/Explanation
Ans:
(a) \( 2^{-5} \) (The denominator is \( 2^5 \), and \( \frac{1}{2^5} = 2^{-5} \))
(b)(i) t = 12 (Using laws of indices: 18 – t = 6 → t = 12)
(b)(ii) \( 48w^{15} \) (Multiply coefficients: 8×6=48; add exponents: 10+5=15)
(a) Find the value of \(137^{0}\).
(b) \(7^{12}\div 7^{P}=7^{17}\)
Find the value of P.
▶️ Answer/Explanation
(a) Ans: 1
Any non-zero number raised to the power of 0 is always 1. This is a fundamental exponent rule.
Thus, \(137^{0} = 1\).
(b) Ans: \(-5\)
Using the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\), we rewrite the equation:
\(7^{12-P} = 7^{17}\).
Since the bases are equal, the exponents must be equal: \(12 – P = 17\).
Solving for \(P\) gives \(P = 12 – 17 = -5\).