NBT Mathematics: Exponents, Surds and Equations Flashcards

Simplify: $\dfrac{9^{n+1} - 9^n}{9^n \cdot 8}$

Answer: $1$

Numerator: $9^n(9-1) = 8 \cdot 9^n$; dividing by $8 \cdot 9^n$ gives 1.

For which value of $x$ is $\sqrt{2x-4}$ defined?

Answer: $x \geq 2$

The expression under the square root must be non-negative: $2x-4 \geq 0 \Rightarrow x \geq 2$.

Solve for $x$: $4^x - 6 \cdot 2^x + 8 = 0$

Answer: $x = 1$ or $x = 2$

Let $k=2^x$: $k^2-6k+8=0 \Rightarrow (k-2)(k-4)=0$; $2^x=2 \Rightarrow x=1$ or $2^x=4 \Rightarrow x=2$.

Simplify: $(x^{-2}y^3)^{-2}$

Answer: $x^4 y^{-6}$

Apply the power: $x^{(-2)(-2)} y^{3(-2)} = x^4 y^{-6}$.

Solve: $x^2 + 2x - 8 = 0$

Answer: $x = 2$ or $x = -4$

$(x+4)(x-2)=0$, so $x=-4$ or $x=2$; thus $x=2$ or $x=-4$.

What is $(-32)^{3/5}$?

Answer: $-8$

$\sqrt[5]{-32} = -2$; then $(-2)^3 = -8$.

Simplify: $\sqrt{18} \times \sqrt{8}$

Answer: $12$

$\sqrt{18 \times 8} = \sqrt{144} = 12$.