What is a square root of a number?

A square root of a number is a value that, when squared , gives the original number . E.g. 3 is the square root of 9 because 3 2 = 3 x 3 = 9.

True or False? Cube roots of negative numbers do not exist .

False. The cube root of a negative number will also be a negative number . For example, -5 is a cube root of -125.

GCSEMathsEdexcelFoundationFlashcardsNumberPowers, Roots & Standard Form

Powers, Roots & Standard Form (Edexcel GCSE Maths: Foundation): Flashcards

Exam code: 1MA1

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What is a square root of a number?

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What is a square root of a number?
A square root of a number is a value that, when squared, gives the original number.
E.g. 3 is the square root of 9 because 3² = 3 x 3 = 9.
True or False?
Every positive number has two square roots.
True.
Every positive number has two square roots, one positive and one negative.
E.g. $\sqrt{64} = 8$ and $- 8$
True or False?
Cube roots of negative numbers do not exist.
False.
The cube root of a negative number will also be a negative number.
For example, -5 is a cube root of -125.
Define the term reciprocal.
The reciprocal of a number is the number that you multiply it by to get 1.
For example, $\frac{2}{3}$ is the reciprocal of $\frac{3}{2}$ .
How do I estimate a root, e.g. the square root of 27?
You can estimate the root of a number by finding the closest integer roots either side of it.
E.g. To find $\sqrt{27}$
$\sqrt{25} = 5$ and $\sqrt{36} = 6$
So $\sqrt{27}$ is between $5$ and $6$
Complete the two index laws below by filling in the missing indices:
$a^{m} \times a^{n} = a^{______}$
$a^{m} \div a^{n} = a^{______}$
The completed laws are:
$a^{m} \times a^{n} = a^{m + n}$
$a^{m} \div a^{n} = a^{m - n}$
Multiplying adds the indices, and dividing subtracts them.
How do you simplify a power that is raised to another power, such as ${(14^{3})}^{2}$ ?
Multiply the two indices: ${(a^{m})}^{n} = a^{m n}$ .
So ${(14^{3})}^{2} = 14^{3 \times 2} = 14^{6}$ .
What is the value of a number raised to the power of $0$ ?
Any non-zero number raised to the power of $0$ is $1$ .
The base makes no difference: $8^{0} = 1$ , $100^{0} = 1$ and ${(- 3)}^{0} = 1$ .
What does a negative index tell you to do, for example in $2^{- 4}$ ?
A negative index means the reciprocal of the positive power: $a^{- n} = \frac{1}{a^{n}}$ .
So $2^{- 4} = \frac{1}{2^{4}} = \frac{1}{16}$ .
How do you expand a product or a fraction that is raised to a power?
Apply the power to each number separately: ${(a b)}^{n} = a^{n} b^{n}$ and ${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$ .
For example, ${(\frac{3}{4})}^{2} = \frac{3^{2}}{4^{2}} = \frac{9}{16}$ .
True or False?
The index laws can be used to simplify $2^{5} \times 4^{3}$ .
True.
Although the two bases look different, $4 = 2^{2}$ , so the whole expression can be written with base $2$ :
$2^{5} \times {(2^{2})}^{3} = 2^{5} \times 2^{6} = 2^{11}$
The index laws only work once the bases are the same.
How do you find the value of $x$ in the equation $6^{10 + x} = 6^{2}$ ?
Since both sides have the same base, their indices must be equal, so $10 + x = 2$ .
Subtracting $10$ from both sides gives $x = - 8$ .
What does a number written in standard form look like?
A number in standard form looks like $a \times 10^{n}$ , e.g. $5.2 \times 10^{3}$ .
It is a number between 1 and 10 multiplied by a power of ten.
When a number is written in standard form $a \times 10^{n}$ , which values can $a$ be?
For $a \times 10^{n}$ , $a$ is at least 1 and less than 10, $1 \leq a < 10$ .
This means there is one non-zero digit before the decimal point.
True or False?
When a number is written in standard form $a \times 10^{n}$ , $n$ can be a fraction.
False.
For a number written in standard form, $a \times 10^{n}$ , the value $n$ must be an integer.
It can be zero, positive or negative.
When a value between 0 and 1, is written in standard form $a \times 10^{n}$ , what type of number will $n$ be?
If a value is between 0 and 1 then when it is written in standard form, $n$ will be a negative integer.
E.g. 0.083 would be written as 8.3 x 10^-2.
How do you work out $(3 \times 10^{8}) \times (2 \times 10^{- 3})$ ?
Multiply the number parts, then add the powers of ten: $3 \times 2 = 6$ and $10^{8} \times 10^{- 3} = 10^{5}$ .
This gives $6 \times 10^{5}$ .
Complete the working by filling in the missing number and the missing index:
$(8 \times 10^{7}) \div (2 \times 10^{3}) =______\times 10^{______}$
The completed working is:
$(8 \times 10^{7}) \div (2 \times 10^{3}) = 4 \times 10^{4}$
Divide the number parts, and subtract the powers of ten.
True or False?
When you multiply two numbers in standard form, the result is automatically in standard form.
False.
For example, $(5 \times 10^{3}) \times (4 \times 10^{2})$ comes out as $20 \times 10^{5}$ , which needs rewriting before it counts as standard form.
A calculation gives the answer $243 \times 10^{20}$ . How do you rewrite this in standard form?
Write $243$ itself in standard form, then combine the two powers of ten:
$243 \times 10^{20} = (2.43 \times 10^{2}) \times 10^{20} = 2.43 \times 10^{22}$
Why should each number in standard form be put in brackets when you type a calculation into a calculator?
A number such as $3 \times 10^{8}$ is a product, not a single value, so without brackets the calculator applies the next operation to only part of it.
For example, typing $6 \times 10^{5} \div 2 \times 10^{3}$ divides by $2$ and then multiplies by $10^{3}$ , instead of dividing by the whole of $2 \times 10^{3}$ .

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