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.

What number do you get when you raise any non-zero number to the power of zero , e.g. 2 0 ?

Any non-zero number raised to the power of 0 is equal to 1 . E.g. 2 0 = 1.

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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 $6$
What number do you get when you raise any non-zero number to the power of zero, e.g. 2⁰?
Any non-zero number raised to the power of 0 is equal to 1.
E.g. 2⁰ = 1.
True or False?
If you raise a non-zero number to the power of 1, you get 1.
False.
Any number raised to the power 1 is just itself.
E.g. $6^{1} = 6$ .
What do you get if you raise a non-zero number to the power of -1,
e.g. 3^-1?
If you raise a non-zero number to the power of -1 you get the reciprocal of the number.
E.g. $3^{- 1} = \frac{1}{3}$ .
What do you get if you raise a positive number to the power of ½,
e.g. 5^1/2?
If you raise a non-zero number to the power of ½ you get its positive square root.
E.g. $5^{\frac{1}{2}} = \sqrt{5}$ .
What is the index law for $a^{m} \times a^{n}$ ?
$a^{m} \times a^{n} = a^{m + n}$
If you multiply two powers with the same base number, you add the indices together.
What is the index law for $a^{m} \div a^{n}$ ?
$a^{m} \div a^{n} = a^{m - n}$
If you divide two powers with the same base number, you subtract one index from the other.
What is the index law for ${(a^{m})}^{n}$ ?
${(a^{m})}^{n} = a^{m \times n}$
If you raise a power to another power, you multiply the indices.
True or False?
$a^{\frac{m}{n}} = (\sqrt[m]{a^{n}})$
False.
$a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$
The denominator of an index is the root and the numerator of an index is the power.
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 multiply two numbers that are written in standard form and give the answer in standard form using index laws?
E.g. $(2 \times 10^{3}) \times (6 \times 10^{5})$
Multiply together the values of $a$ from both numbers and write this in standard form if needed, e.g. $2 \times 6 = 12 = 1.2 \times 10^{1}$ .
Use index laws to combine the powers of 10, e.g. $10^{3} \times 10^{5} = 10^{8}$ .
Multiply the two parts together, e.g. $1.2 \times 10^{1} \times 10^{8}$ .
Simplify if necessary, using index laws again e.g. $1.2 \times 10^{9}$ .
How do you divide two numbers that are written in standard form and give the answer in standard form using index laws?
E.g. $(2 \times 10^{9}) \div (8 \times 10^{2})$
Divide the values of $a$ for both numbers and write this in standard form if needed, e.g. $2 \div 8 = 0.25 = 2.5 \times 10^{- 1}$
Combine the powers of 10 using index laws, e.g. $10^{9} \div 10^{2} = 10^{7}$
Multiply the two parts together, e.g. $2.5 \times 10^{- 1} \times 10^{7}$
Simplify using index laws again if necessary, e.g. $2.5 \times 10^{6}$

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