Variables with Exponents

An exponent tells you how a power is formed. For a positive integer exponent, \(x^n\) is a product of n factors, each equal to x. Thus, \(x^4=x\cdot x\cdot x\cdot x\); it does not mean \(4x\).

A variable stands for a value that may be unknown or allowed to vary. Exponent rules help us evaluate and simplify expressions without writing out every factor.

Substitute values before calculating

Evaluate \(3x^2-2y\) when \(x=-2\) and \(y=5\):

\[3(-2)^2-2(5)=3(4)-10=2.\]

Put a substituted negative number in parentheses. The expressions \((-2)^2=4\) and \(-2^2=-4\) are different: in the second expression, the exponent is evaluated before the leading minus sign.

Follow the order of operations: grouping, exponents, multiplication and division from left to right, then addition and subtraction from left to right.

Adding and subtracting like terms

Like terms have the same variable factors raised to the same powers. Add or subtract their coefficients:

\[3x^2+4x^2=7x^2.\]

The exponent stays the same. The terms \(3x^2\) and \(4x\) are not like terms and cannot be combined into a single monomial by addition.

Multiplying powers with the same base

Add the exponents when multiplying powers of the same base:

\[x^m x^n=x^{m+n}.\]

For example,

\[(3x^2y^2)(4x^2)=12x^4y^2.\]

Multiply the coefficients \(3\times4\), then combine the x-factors using \(x^2x^2=x^4\). The factor \(y^2\) remains unchanged. This is multiplication of monomials, not addition of like terms.

Dividing powers and raising a power to a power

For a nonzero base, subtract exponents when dividing:

\[\frac{x^m}{x^n}=x^{m-n}\qquad(x\ne0).\]

For example, \(x^7/x^3=x^4\) when \(x\ne0\).

When raising a power to another integer power, multiply the exponents:

\[(x^m)^n=x^{mn}.\]

Thus, \((x^3)^2=x^6\). A power applies to every factor inside a product:

\[(2xy^2)^3=8x^3y^6.\]

It does not distribute over addition: \((x+y)^2=x^2+2xy+y^2\), not generally \(x^2+y^2\).

Zero and negative exponents

For \(x\ne0\),

\[x^0=1,\qquad x^{-n}=\frac1{x^n}.\]

For example, \(2^{-3}=1/8\). A negative exponent means a reciprocal, not a negative result. Expressions such as \(0^{-1}\) are undefined because division by zero is undefined. The expression \(0^0\) needs context and should not be assigned a value by applying the nonzero-base rule. For powers of ten, compare decimal and exponent forms with the scientific notation calculator.

Substituting one expression for another

If \(y=2x-5\), then substituting into \(3x-4y=7\) gives

\[3x-4(2x-5)=7.\]

Distribute the \(-4\) before combining terms:

\[3x-8x+20=7,\qquad-5x=-13,\qquad x=\frac{13}{5}.\]

Then \(y=2(13/5)-5=1/5\). Checking in the original equation gives \(3(13/5)-4(1/5)=7\).

Replace the entire variable with the entire expression. Dropping parentheses or changing the equals signs into unrelated steps can turn a valid substitution into a false statement.

Fractional exponents need domain checks

For \(x\ge0\), \(x^{1/2}=\sqrt{x}\). More general fractional powers require care with signs and the chosen number system. Rules that work for integer powers cannot always be extended to arbitrary real powers without restrictions.

Start by identifying the base, the exponent, and the allowed values. That habit makes both simplification and substitution more reliable.