NBT Mathematics: Exponents, Surds and Equations Flashcards

Solve for $x$ and $y$: $2x + 3y = 12$ and $x = y + 1$

Answer: $x = 3, y = 2$

Substituting $x = y+1$: $2(y+1)+3y=12 \Rightarrow 5y=10 \Rightarrow y=2$, so $x=3$.

Which of the following is equivalent to $a^{1/2} \cdot a^{3/2}$?

Answer: $a^2$

$a^{1/2+3/2} = a^{4/2} = a^2$.

Rationalise and simplify: $\dfrac{\sqrt{3}}{\sqrt{3}+1}$

Answer: $\dfrac{3-\sqrt{3}}{2}$

Multiply by $\dfrac{\sqrt{3}-1}{\sqrt{3}-1}$: $\dfrac{\sqrt{3}(\sqrt{3}-1)}{3-1} = \dfrac{3-\sqrt{3}}{2}$.

Solve: $x^2 - 4 = 0$

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

$x^2 = 4 \Rightarrow x = \pm 2$.

Simplify: $3\sqrt{8} - \sqrt{2}$

Answer: $5\sqrt{2}$

$3\sqrt{8} = 3 \cdot 2\sqrt{2} = 6\sqrt{2}$, so $6\sqrt{2} - \sqrt{2} = 5\sqrt{2}$.

If $9^x = 3^{x+4}$, what is $x$?

Answer: $x = 4$

$9^x = 3^{2x}$; equating: $2x = x + 4 \Rightarrow x = 4$.

Which is the correct factorised form of $x^2 - 9$?

Answer: $(x-3)(x+3)$

$x^2 - 9$ is a difference of two squares: $(x-3)(x+3)$.