write the laws of indices
Answers
- Step-by-step explanation
The first law: multiplication. If the two terms have the same base (in this case. ...
The second law: division. If the two terms have the same base (in this case. ...
The third law: brackets. ...
Negative powers. ...
Power of zero. ...
Fractional powers.
example: Index laws are the rules for simplifying expressions involving powers of the same base number. = ( 3 √ 27)2 = (3)2 = 9. (2) Watch out for powers of negative numbers. For example, (−2)3 = −8 and (−2)4 = 16, so (−x)5 = −x5 and (−x)6 = x6.
Step-by-step explanation:
The first law: multiplication
The first law: multiplicationIf the two terms have the same base (in this case x) and are to be multiplied together their indices are added.
The first law: multiplicationIf the two terms have the same base (in this case x) and are to be multiplied together their indices are added.In general: x^m \times x^n = x^{m+n}
The second law: division
The second law: divisionIf the two terms have the same base (in this case x) and are to be divided their indices are subtracted.
The second law: divisionIf the two terms have the same base (in this case x) and are to be divided their indices are subtracted.In general: \dfrac{x^m}{x^n}=x^{m-n}
The third law: brackets
The third law: bracketsIf a term with a power is itself raised to a power then the powers are multiplied together.
The third law: bracketsIf a term with a power is itself raised to a power then the powers are multiplied together.In general: (x^m)^n = x^{m \times n}
Negative powers
Negative powersConsider this example: \dfrac{a^2}{a^6} = a^{2-6} = a^{-4}
Negative powersConsider this example: \dfrac{a^2}{a^6} = a^{2-6} = a^{-4}Also we can show that: \dfrac{a^2}{a^6} = \dfrac{1}{a^4}
Negative powersConsider this example: \dfrac{a^2}{a^6} = a^{2-6} = a^{-4}Also we can show that: \dfrac{a^2}{a^6} = \dfrac{1}{a^4}So a negative power can be written as a fraction.
Negative powersConsider this example: \dfrac{a^2}{a^6} = a^{2-6} = a^{-4}Also we can show that: \dfrac{a^2}{a^6} = \dfrac{1}{a^4}So a negative power can be written as a fraction.In general: x^{-m} = \dfrac{1}{x^m}
Power of zero
Power of zeroThe second law of indices helps to explain why anything to the power of zero is equal to one.
Power of zeroThe second law of indices helps to explain why anything to the power of zero is equal to one.We know that anything divided by itself is equal to one. So \dfrac {x^3}{x^3} = 1
Power of zeroThe second law of indices helps to explain why anything to the power of zero is equal to one.We know that anything divided by itself is equal to one. So \dfrac {x^3}{x^3} = 1Also we know that \dfrac {x^3}{x^3} = x^{3-3} = x^0 = 1
Power of zeroThe second law of indices helps to explain why anything to the power of zero is equal to one.We know that anything divided by itself is equal to one. So \dfrac {x^3}{x^3} = 1Also we know that \dfrac {x^3}{x^3} = x^{3-3} = x^0 = 1Therefore, we have shown that \dfrac {x^3}{x^3} = x^0 = 1
Fractional powers
Fractional powersBoth the numerator and denominator of a fractional power have meaning.
Fractional powersBoth the numerator and denominator of a fractional power have meaning.The bottom of the fraction stands for the type of root; for example, x^{\frac{1}{3}} denotes a cube root \sqrt[3]{x}
Fractional powersBoth the numerator and denominator of a fractional power have meaning.The bottom of the fraction stands for the type of root; for example, x^{\frac{1}{3}} denotes a cube root \sqrt[3]{x}The top line of the fractional power gives the usual power of the whole term.
Fractional powersBoth the numerator and denominator of a fractional power have meaning.The bottom of the fraction stands for the type of root; for example, x^{\frac{1}{3}} denotes a cube root \sqrt[3]{x}The top line of the fractional power gives the usual power of the whole term.For example: x^{\frac{2}{3}} = (\sqrt[3]{x})^2
In general: x^{\frac{m}{n}} = (\sqrt[n]{x})^m
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