Math Tests - mathematics practice questions

Numbers 1-12 squared

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Square numbers

A square number is a number multiplied by itself. The symbol is 2 (we call it exponent). The number squared is called the base:

$$2^2= 2 × 2 = 4$$
$$3^2 = 3 × 3 = 9$$
$$4^2 = 4 × 4 = 16$$

We can read this as two/three/four to the second power.

A cube number is a number multiplied by itself 3 times:

$$2^3 = 2 × 2 × 2 = 8$$
$$3^3 = 3 × 3 × 3 = 27$$


Square roots

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Square roots

The root of a number y is another number, which when multiplied by itself a given number of times, equals y. The second root (also called the square root) of 16 is 4:

$$16=4×4$$
We write this as:

$$√16=4$$
or
$$√^2{16}=4$$

For every even-degree root (2nd, 4th, 6th ....) there are two roots. This is because multiplying two positive or two negative numbers both produce a positive result. E.g:

$$4×4=16$$
$$(−4)×(−4)=16$$

Thus, $√^2{16}=±4$

The value inside the symbol √(called radical) is called radicand. For $√^3{27};;;$ 27 is radicand and 3 is the index.



Exponents 0, 1 and higher

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Exponents

The exponent of a number x says how many times to multiply the number x by itself. Number x is called the base.

$$2^4=2×2×2×2=16$$
$$3^3=3×3×3=81$$

There are some special exponents we need to remember:

$$x^0=1$$
$$x^1=x$$
$$x^{-n}=1/x^n$$
$$x^{1/n}=√^n{x}$$


Exponents and roots with decimal base

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Exponents and roots with fractional base

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$$a^{−1}=1/a$$

$$a^{−n}=1/{a^n}$$

$$a^{1/n}=√^{n}a$$

$${a^m}/{a^n}=a^{m−n}$$

$$√{a/b} = √a/√b$$

$$(a/b)^n=a^n/b^n$$


Simplify square root

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Simplifying square roots

To simplify roots you can use Product Property of Square Roots:

$$√{a×b}=√a×√b$$

At first, find the perfect square inside the square root:

$$√24;→;24=2×2×3×2=4×6$$
$$√24=√{4×6}=√4×√6=2×√6$$

This can be written as 2√6



Numerical expressions with exponents

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$$a^2=a×a$$

$$a^1=a$$

$$a^0=1$$

$$a^{−1}=1/a$$

$$a^{−n}=1/{a^n}$$

$$a^{1/n}=√^{n}a$$

$$a^m×a^n=a^m a^n=a^{(m+n)}$$

$${a^m}/{a^n}=a^{m−n}$$

$$(a^m)^n=a^{(m×n)}$$

$$√^n{a}√^n{b} = √^n{ab}$$

$$√{ab} = √a×√b$$

$$(a/b)^2=a^2/b^2$$


Numerical expressions with roots and exponents

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Simplifying square roots

To simplify roots you can use Product Property of Square Roots:

$$√{a×b}=√a×√b$$

At first, find the perfect square inside the square root:

$$√24;→;24=2×2×3×2=4×6$$
$$√24=√{4×6}=√4×√6=2×√6$$

This can be written as 2√6

You can also use these formulae:

$$a^2=a×a$$

$$a^1=a$$

$$a^0=1$$

$$a^{−1}=1/a$$

$$a^{−n}=1/{a^n}$$

$$a^m×a^n=a^m a^n=a^{(m+n)}$$

$${a^m}/{a^n}=a^{m−n}$$

$$(a^m)^n=a^{(m×n)}$$

$$a^{1/n}=√^{n}a$$

$$√^n{a}√^n{b} = √^n{ab}$$

$$√{ab} = √a×√b$$

$$√{a/b}=√a/√b$$

$$(a/b)^2=a^2/b^2$$


   
   

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