Flashcards · 12 cards

Indices & standard form

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Rule
$a^m \times a^n = \,?$
Answer
$a^{m + n}$ : add the indices. This works only when the base is the same.
MultiplyingCount the a's: add the indices
From the lesson: The laws of indices
Rule
$a^m \div a^n = \,?$
Answer
$a^{m - n}$ : subtract the indices.
DividingCancel the a's: subtract the indices
From the lesson: The laws of indices
Rule
$(a^m)^n = \,?$
Answer
$a^{mn}$ : multiply the indices.
A power of a powerGroups of a's: multiply the indices
From the lesson: The laws of indices
Rule
What are $a^0$ and $a^{-n}$ ?
Answer
$a^0 = 1$ and $a^{-n} = \dfrac{1}{a^n}$ . So $2^{-3} = \dfrac18$ : small and positive, not $-8$ .
Zero and negativeKeep halving: 2⁰ = 1, 2⁻¹ = ½
From the lesson: The laws of indices
Rule
What does $a^{\frac mn}$ mean?
Answer
Root first, then power: $a^{\frac mn} = \left(\sqrt[n]{a}\right)^m$ . So $8^{\frac23} = 2^2 = 4$ .
Root first, then powerThe bottom of the fraction is the root; the top is the power
From the lesson: The laws of indices
Know it
$\left(\dfrac ab\right)^{-n} = \,?$
Answer
$\left(\dfrac ba\right)^{n}$ : turn the fraction over. So $\left(\frac23\right)^{-2} = \frac94$ .
From the lesson: The laws of indices
Know it
Can you simplify $2^3 \times 3^4$ by adding the indices?
Answer
No. The laws work only on the same base. Where you can, change the bases to match first, such as $8 = 2^3$ or $27 = 3^3$ .
From the lesson: The laws of indices
Know it
How do you solve an equation like $27^{2x - 1} = 9^{x + 3}$ ?
Answer
Write both sides as powers of the same base (here 3), then set the indices equal: $3(2x - 1) = 2(x + 3)$ . Bracket the index before you multiply.
From the lesson: Equations with indices
Rule
How do you solve $2^{2x} - 6 \times 2^x + 8 = 0$ ?
Answer
Let $y = 2^x$ , so $2^{2x} = y^2$ : $y^2 - 6y + 8 = 0$ gives $y = 2$ or $4$ . Then $2^x = 2$ or $2^x = 4$ , so $x = 1$ or $x = 2$ .
A quadratic in disguiseLet y = 2ˣ
From the lesson: Equations with indices
Rule
What is standard form?
Answer
$A \times 10^n$ with $1 \le A < 10$ and $n$ a whole number. For example $34\,500\,000 = 3.45 \times 10^7$ .
Moving the pointEach jump right makes the number 10 times bigger, so the power of 10 goes down by 1 to keep the value
From the lesson: Standard form
Know it
How do you multiply two numbers in standard form?
Answer
Multiply the numbers in front, add the powers of 10, then put it back in standard form: $(2.5 \times 10^6)(6.4 \times 10^{-9}) = 16 \times 10^{-3} = 1.6 \times 10^{-2}$ .
From the lesson: Standard form
Which method?
NECO 2023 · Paper 2 · Q4 (a)
Solve the equation $27^{2x - 1} \times \left(\frac13\right)^{-(3x + 2)} = 9^{x + 3}$ .
What is the first move?
Answer
Write every number as a power of 3: $27 = 3^3$ , $\frac13 = 3^{-1}$ , $9 = 3^2$ . Add the indices on the left, then set the two indices equal.
From the lesson: Equations with indices Try the question