A negative exponent indicates the reciprocal of the base raised to the absolute value of that exponent. This means that for any non-zero number 'a', the expression 'a^{-n}' is equal to '1/a^{n}', where 'n' is a positive integer. Understanding negative exponents is essential for simplifying expressions, especially in algebra, as they can help clarify relationships between different powers of numbers.
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Negative exponents provide a way to express division in terms of multiplication, making calculations simpler.
When dealing with multiple bases, each with a negative exponent, you can apply the rule to convert each one into a fraction.
Negative exponents are often encountered in scientific notation, allowing for compact representation of very large or small numbers.
When multiplying numbers with negative exponents, you can add the exponents once they are converted to their positive forms.
It's crucial to remember that zero raised to any negative exponent is undefined, as it involves division by zero.
Review Questions
How do you simplify an expression involving a negative exponent, such as 'x^{-3}'?
'x^{-3}' can be simplified by applying the rule for negative exponents. You rewrite it as '1/x^{3}', which expresses the original term in terms of its positive exponent. This transformation highlights the reciprocal nature of negative exponents and allows for easier manipulation in equations or further simplification.
What happens when you multiply two numbers with negative exponents, for example, 'a^{-m} * b^{-n}'?
When multiplying two numbers with negative exponents like 'a^{-m} * b^{-n}', you first convert them into their positive forms: this gives you '(1/a^{m}) * (1/b^{n})'. This results in '1/(a^{m} * b^{n})', demonstrating how negative exponents allow us to express division through multiplication.
Evaluate and explain the expression '4^{-2} * 2^{-3}' and discuss its significance.
'4^{-2} * 2^{-3}' can be evaluated by first rewriting it: '1/(4^{2}) * 1/(2^{3})'. This becomes '1/(16) * 1/(8)', which equals '1/128'. The significance here lies in understanding how negative exponents represent division and simplify complex expressions into more manageable forms. This illustrates the power of exponents in mathematical operations and their utility in various contexts.
The mathematical operation of raising one number to the power of another, represented as 'a^b', where 'a' is the base and 'b' is the exponent.
reciprocal: The reciprocal of a number 'x' is '1/x'. It is used in conjunction with negative exponents to express the relationship between a base and its negative exponent.
power of a power: An exponent rule that states when raising a power to another power, you multiply the exponents, expressed as '(a^m)^n = a^{m*n}'.