National 5MathsSQAFlashcardsNumerical SkillsLaws of Indices

Laws of Indices (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontLaws of Indices
Complete the three index laws by filling in the missing indices.
$a^{m} \times a^{n} = a^{______}$
$a^{m} \div a^{n} = a^{______}$
${(a^{m})}^{n} = a^{______}$

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Complete the three index laws by filling in the missing indices.
$a^{m} \times a^{n} = a^{______}$
$a^{m} \div a^{n} = a^{______}$
${(a^{m})}^{n} = a^{______}$
The completed laws are:
$a^{m} \times a^{n} = a^{m + n}$
$a^{m} \div a^{n} = a^{m - n}$
${(a^{m})}^{n} = a^{m n}$
All three need the same base, which is why the base never changes.
Which index law explains why $a^{0}$ is equal to 1?
Dividing a power by itself gives $a^{m} \div a^{m} = a^{m - m} = a^{0}$ , and any non-zero number divided by itself is 1.
So $a^{0} = 1$ for every non-zero value of $a$ .
What does a negative index such as $a^{- 3}$ mean?
A negative index means the reciprocal of the positive power, so $a^{- n} = \frac{1}{a^{n}}$ .
For example $a^{- 3} = \frac{1}{a^{3}}$ , and $a^{- 1} = \frac{1}{a}$ .
True or False?
$5^{- 2}$ is a negative number.
False.
$5^{- 2} = \frac{1}{5^{2}} = \frac{1}{25}$ , which is positive.
A negative index makes the value smaller, but it never changes its sign.
What does a fractional index such as $a^{\frac{2}{3}}$ mean?
The denominator of the fraction gives a root and the numerator gives a power, so $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ .
For $a^{\frac{2}{3}}$ that is ${(\sqrt[3]{a})}^{2}$ , which is also equal to $\sqrt[3]{a^{2}}$ .
Complete the rule for a fraction raised to a negative index.
${(\frac{a}{b})}^{- n} = {(\frac{______}{______})}^{n}$
The completed rule is:
${(\frac{a}{b})}^{- n} = {(\frac{b}{a})}^{n} = \frac{b^{n}}{a^{n}}$
A negative index turns the fraction upside down.
Evaluate $8^{\frac{2}{3}}$ without a calculator.
Take the root first and then the power, so $8^{\frac{2}{3}} = {(\sqrt[3]{8})}^{2} = 2^{2} = 4$ .
Doing the power first gives $\sqrt[3]{64}$ , which reaches the same answer but leaves a much larger number to work with.
True or False?
Two powers with different bases can be combined into one, provided they have the same index.
True.
Raising a product to a power applies the power to each factor, so ${(a b)}^{m} = a^{m} b^{m}$ , and that rule can be used in reverse.
For example $2^{4} \times 3^{4} = 6^{4}$ .
How do you simplify a product such as $(3 x^{7}) \times (6 x^{4})$ into a single term?
Deal with the number parts and the algebra parts separately.
Multiplying the numbers gives $3 \times 6 = 18$ and adding the indices gives $x^{7 + 4} = x^{11}$ , so the answer is $18 x^{11}$ .
If both sides of an equation are powers of the same base, what can you say about the two indices?
The two indices must be equal, which turns the equation into one with no powers in it.
For example $4^{3 x} = 4^{9}$ gives $3 x = 9$ , so $x = 3$ .

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