CSEE (Form 4) Exponents, Radicals and Logarithms — practice questions

1. Rewrite both sides to a common base of 2: $8=2^3$ and $16=2^4$. 2. The left side becomes $2^{3(x-1)}$ and the right side $2^{4}$. 3. With equal bases the exponents must match: $3(x-1)=4$. 4. Expanding gives $3x-3=4$, so $3x=7$. 5. Therefore $x=\dfrac73$. Failing to bring both sides to the same base is what blocked most candidates here. _Source: NECTA CSEE 2022 Basic Mathematics 041, Question 2: Exponents, Radicals and Logarithms_

1. Use $\log_a M+\log_a N=\log_a(MN)$ to combine the two terms. 2. That gives $\log_a\left(\sqrt{a}\cdot a^2\right)$. 3. Write the radical as a power: $\sqrt{a}=a^{1/2}$, so the product is $a^{1/2+2}=a^{5/2}$. 4. Then $\log_a\left(a^{5/2}\right)=\dfrac52$, since $\log_a(a^k)=k$. 5. Answering $2$ drops the contribution of the square-root term entirely. _Source: NECTA CSEE 2022 Basic Mathematics 041, Question 2: Exponents, Radicals and Logarithms_

1. Rationalising removes the radicals from the denominator. 2. The denominator is $\sqrt6-\sqrt2$, so multiply above and below by its conjugate. 3. The conjugate flips the middle sign, giving $\sqrt6+\sqrt2$. 4. Then $(\sqrt6-\sqrt2)(\sqrt6+\sqrt2)=6-2=4$, which is rational. 5. Using $\sqrt6-\sqrt2$ again, a recorded error, leaves a radical behind instead of clearing it. _Source: NECTA CSEE 2022 Basic Mathematics 041, Question 2: Exponents, Radicals and Logarithms_