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Longhand

Before the midterm

Misapplying a law of exponents

The habit: A power of a power multiplies the exponents, a product adds them, a quotient subtracts them, and an nth root is the power 1/n.

3 practice questions are built to catch it, in 2 topics.

Practise against it: 2 questions

What it looks like

A question from functions, graphs and inverses, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

For x>0x > 0, the expression below equals a single power xkx^{k}. Find kk.

(x4)3x5x\frac{\left(x^{4}\right)^{3}}{x^{5}\sqrt{x}}
With the slip
1.5\displaystyle 1.5
Worked through
k=132\displaystyle k = \frac{13}{2}

(x4)³ is x4 multiplied by itself three times, x¹², not x7: a power of a power multiplies the exponents.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    Find the exact value of the expression below, where log⁡\log is the natural logarithm. log⁡ ⁣(e5e3)+2log⁡e\displaystyle \log\!\left(\frac{e^{5}}{\sqrt[3]{e}}\right) + 2\log\sqrt{e}

    Functions, graphs and inversesIntroductory

  2. 02
    From the definition of the derivative, ddxax=C(a) ax\dfrac{d}{dx}a^{x} = C(a)\,a^{x} for every base a>0a > 0, where C(a)=lim⁡h→0ah−1hC(a) = \displaystyle\lim_{h \to 0} \dfrac{a^{h} - 1}{h}. A calculator estimates C(3)≈1.0986C(3) \approx 1.0986. Without estimating another limit, find the slope of y=9xy = 9^{x} at x=12x = \tfrac{1}{2}.

    Exponential and trigonometric functionsStandard

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