NBT NBT Mathematics: Exponents, Surds and Equations 5

Question 1

Simplify: $\left(\dfrac{x^4}{y^2}\right)^{-1/2}$

Accepted Answer

$\dfrac{y}{x^2}$

Answer

Applying the negative exponent inverts the fraction and the half-power: $\dfrac{y}{x^2}$.

Question 2

Solve for $x$: $2^{2x} - 5 \cdot 2^x + 4 = 0$

Accepted Answer

$x = 0$ or $x = 2$

Answer

Let $k = 2^x$: $k^2 - 5k + 4 = 0 \Rightarrow (k-1)(k-4) = 0$, so $k=1$ or $k=4$, giving $x=0$ or $x=2$.

Question 3

What is $\sqrt[3]{-64}$?

Accepted Answer

$-4$

Answer

$(-4)^3 = -64$, so $\sqrt[3]{-64} = -4$.

Question 4

Solve for $x$: $\sqrt{x+1} = x - 1$

Accepted Answer

$x = 3$

Answer

Squaring: $x+1=(x-1)^2=x^2-2x+1$; so $x^2-3x=0 \Rightarrow x(x-3)=0$; $x=0$ fails check (LHS=1, RHS=-1), so $x=3$.

Question 5

Simplify: $2\sqrt{3} 	imes 3\sqrt{3}$

Accepted Answer

$18$

Answer

$2 	imes 3 	imes \sqrt{3} 	imes \sqrt{3} = 6 	imes 3 = 18$.

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