Why isn't 8 minus 2 times 3 equal to 18?

Multiplication is evaluated before subtraction regardless of the order the numbers appear in, so 2 times 3 is calculated first, giving 6, and then 8 minus 6 gives 2. Reading the expression strictly left to right without applying order of operations is what produces the wrong answer of 18.

Why does a negative number have no real square root?

Squaring any real number always produces a non-negative result, since a negative multiplied by a negative is positive, just like a positive multiplied by a positive. There is no real number whose square is negative, so within the real number system, the square root of a negative number is undefined.

What does an exponent of 1/2 actually mean?

It means the square root. More generally, an exponent of 1/n means the nth root of the base, so 1/3 means the cube root and 1/4 means the fourth root. Fractional exponents are a compact way of writing a root as a power.

How can I tell if a fraction will convert to a terminating or a repeating decimal?

Reduce the fraction fully, then look at the denominator. If its only prime factors are 2 and 5, the decimal terminates, like 1/8 becoming 0.125. Any other prime factor in the denominator, like 3, 7, or 11, produces a decimal that repeats forever.

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This category covers general arithmetic and algebra questions: order of operations, exponents, roots, and decimal conversion. These are the rules a calculator applies silently and that produce a different, wrong answer the moment they're applied in the wrong sequence by hand.

Order of operations

Order of operations, often taught as PEMDAS or BODMAS, sets the sequence for evaluating an expression with more than one kind of operation: parentheses or brackets first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication doesn't automatically outrank division, and addition doesn't automatically outrank subtraction; within each pair, the operations are evaluated in the order they appear, left to right, not by which one looks more important.

The expression 8 minus 2 times 3 is not 18. Multiplication happens first regardless of the order the numbers are written in, giving 8 minus 6, which is 2. Getting this wrong is the single biggest source of a 'calculator error' that isn't actually the calculator's fault, since most calculators apply the standard order automatically and a hand calculation that skips it produces a different result.

Negative numbers and square roots

A negative number has no real square root, because squaring any real number, positive or negative, always produces a positive result: negative times negative is positive, and positive times positive is positive. There is no real number that, multiplied by itself, gives a negative answer, which is why the square root of -9 is undefined within the real numbers. Mathematicians handle this by defining an imaginary unit, i, equal to the square root of -1, which extends the number system to cover such cases, a separate topic from everyday arithmetic.

Fractional exponents

A fractional exponent combines a power and a root in one notation. The exponent 1/2 means take the square root; the exponent 1/3 means take the cube root; more generally, x to the power of 1/n means the nth root of x. A number raised to 3/2 means take the square root first and then cube the result, or cube first and then take the square root; both orders give the same answer for a positive base.

Terminating versus recurring decimals

A fraction converts to a terminating decimal, one that ends after a finite number of digits, only when its denominator, once fully reduced, has no prime factors other than 2 and 5, the two prime factors of 10. One quarter, 1/4, terminates as 0.25 because 4 is 2 squared. One third, 1/3, never terminates; it produces 0.333 repeating forever, because 3 is not a factor of any power of 10. Recognizing which fractions terminate predicts whether a decimal conversion rounds cleanly or carries a repeating tail that a calculator display has to truncate somewhere.

Frequently asked questions

Why isn't 8 minus 2 times 3 equal to 18?: Multiplication is evaluated before subtraction regardless of the order the numbers appear in, so 2 times 3 is calculated first, giving 6, and then 8 minus 6 gives 2. Reading the expression strictly left to right without applying order of operations is what produces the wrong answer of 18.
Why does a negative number have no real square root?: Squaring any real number always produces a non-negative result, since a negative multiplied by a negative is positive, just like a positive multiplied by a positive. There is no real number whose square is negative, so within the real number system, the square root of a negative number is undefined.
What does an exponent of 1/2 actually mean?: It means the square root. More generally, an exponent of 1/n means the nth root of the base, so 1/3 means the cube root and 1/4 means the fourth root. Fractional exponents are a compact way of writing a root as a power.
How can I tell if a fraction will convert to a terminating or a repeating decimal?: Reduce the fraction fully, then look at the denominator. If its only prime factors are 2 and 5, the decimal terminates, like 1/8 becoming 0.125. Any other prime factor in the denominator, like 3, 7, or 11, produces a decimal that repeats forever.