  Math Tests - mathematics practice questions

Numbers 1-12 squared ## 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 ## 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 ## 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 fractional base \$\$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 ## 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 \$\$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 ## 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\$\$