Anything To The Zeroth Power
Ability Of 0
Here you lot will learn everything you need to know about raising terms to the power of 0 for GCSE & iGCSE maths (Edexcel, AQA and OCR).
Expect out for the laws of indices worksheets and exam questions at the terminate.
What is raising a value to the ability of 0?
Whatever non-goose egg value raised to the power of 0 is equal to 1.
Laws of indices worksheets (includes power of 0)
Go your free to the power of 0 worksheet of 20+ laws of indices questions and answers. Includes reasoning and applied questions.
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Laws of indices worksheets (includes power of 0)
Get your free to the power of 0 worksheet of 20+ laws of indices questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREE
How to heighten something to the power of 0
Raising a term to the zeroth power means multiplying the term past itself zilch times. This will give i.
Let's look at this in three different means:
1 Division
When we divide something past itself we get 1.
Eastward.g.
\[\begin{aligned} &5 \div 5=1 \\\\ &\frac{5}{six} \div \frac{5}{6}=1 \\\\ &2 ten \div 2 x=1 \stop{aligned}\]
So,
\[x^{2} \div x^{ii}=1\]
Using the division power rule (exponent dominion) when we split up two terms with the same base nosotros subtract the powers.
\[10^{two} \div x^{2}=x^{2-2}=x^{0}\]
So this ways that
\[x^{0}=i\]
2one 10 the base
Another way to think near this is nosotros tin write:
\[ii^{3}=ii \times ii \times 2\]
Which is exactly the same as
\[2^{iii}=1 \times 2 \times 2 \times 2\]
In other words we can think of 23 equally telling us to do 1 multiplied by 2 nonetheless many times the power tells us, in this example three times.
If nosotros continue this design we get the following:
\begin{assortment}{l} ii^{three}=i \times 2 \times 2 \times 2 \\ 2^{2}=i \times ii \times 2 \\ 2^{1}=1 \times ii \\ 2^{0}=1 \end{array}
In other words we can think of 20 as telling us to exercise 1 multiplied by 2 however many times the power tells united states of america, in this case 0 times.
So,
\[two^{0}=1\]
3Comparing other powers of the same base
Nosotros could also think virtually information technology similar this
\[\begin{aligned} 2^{2}&=1 \times two \times ii=iv \\\\ ii^{i}&=1 \times 2=2 \\\\ two^{0}&=1 \\\\ 2^{-i}&=one \times \frac{1}{2}=\frac{1}{2} \\\\ 2^{-2}&=1 \times \frac{1}{ii} \times \frac{i}{2}=\frac{1}{four} \end{aligned}\]
Each time the index power decreases by 1, nosotros dissever the value by whatever the base is.
In this case we separate by 2.
Then,
\[ii^{0}=1\]
Whichever way we look at it, if a term has a aught exponent (it is raised to the ability of zero) its value is 1.
Annotation – Raising 0 to the power of 0 is a complicated problem and its not equally easy every bit maxim the reply is 1.
At that place are many different arguments to this problem and some conclude that the answer should exist i and others say it should be 0. A lot of mathematicians like to classify 0^{0} as an undefined value and its something that gets explored using much more complicated mathematical techniques.
To the power of 0 examples
Example one: no coefficient in front end of base
Simplify:
\[a^{0}\]
= i
Note: Annihilation to the ability of zero is 1.
Case 2: coefficient in forepart of base
Simplify:
\[6a^{0}\]
= 6(1) = six
We can run across that
\[six \times 1=six\]
Multiplying anything past i leaves it unchanged; this is chosen the multiplicative identity.
Example iii: coefficient in front of base and positive powers
Simplify:
\[\frac{2 ten^{0}}{4^{ii}}\]
\[=\frac{ii(i)}{4^{ii}}=\frac{2}{16}=\frac{1}{8}\]
Example 4: with negative exponents and decimals
This example uses negative numbers equally the indices. It is a skilful idea to check out our negative indices page to come across how to change a negative index on the numerator into a positive number on the denominator.
Simplify:
\[2.v x^{0} \div 2^{-2}\]
\[=2.5(one) \div \frac{1}{2^{ii}} \]
\[=2.v \div \frac{1}{four} \]
\[=ii.five \times four \]
\[=ten\]
Common misconceptions
- Disruptive integer and fractional powers
Raising a term to the ability of 2 means we foursquare it
E.g.
\[a^{ii}=a \times a\]
Raising a term to the power of
\[\frac{i}{2}\]
means we find the foursquare root of information technology
E.thousand.
\[a^{\frac{1}{2}}=\pm \sqrt{a}\]
Raising a term to the power of three means nosotros cube it
E.thou
\[a^{3}=a \times a \times a\]
Raising a term to the power of
\[\frac{1}{3}\]
means nosotros detect the cube root of it
E.g.
\[a^{\frac{1}{three}}=\sqrt [3] {a}\]
- Indices, powers or exponents
Indices can also be chosen powers or exponents.
- Raising whatsoever term or real number to the power of 0 is 1.
Regardless of whether information technology is a whole number or a decimal or a fraction, or a positive or a negative number, or a rational number (e.g. 4, 0.25, ½ etc.), or an irrational number (e.g. π, √v, east (Euler's number) etc.) raising a base number or a base variable to the power 0 will give a value of 1.
Raising algebraic polynomials to the power of 0 is also 1.
Whatsoever index that is a non-zero number will non give 1 unless the base of operations value is 1.
Practice to the power of 0 questions
This is because any abiding or variable raised to the power aught is equal to 1
This is because the variable, x , raised to the power zero equals 1. Therefore, nosotros have 8 lots of 1 , which is 8 .
This is considering the variable, x , raised to the power zero equals 1. The coefficient, 2^2 is equal to iv . Therefore, nosotros have 4 lots of 1 , which is iv .
To the power of 0 GCSE questions
ane. Work out the value of
8 ^ {0}
(1 marking)
Evidence reply
2. Simplify
\frac { x ^ {0} \times x ^ {five} }{ x ^ {v} \times x ^ {-5} }
(2 marks)
Evidence reply
x^{0}=1 \text \quad { and } \quad ten^{five} \times x^{-v}=10^{0}=1 \quad(1)
\frac {1 \times ten ^ {5} } {i} = x ^ {five}\quad(1)
(2)
three. Simplify
\left(\left(four^{\frac{1}{2}}\right)^{iii}\right)^{0}
(1 marker)
Show respond
Learning checklist
You have now learned how to:
- Simplify expressions involving the laws of indices
- Summate with roots, and with integer and partial indices
Withal stuck?
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Anything To The Zeroth Power,
Source: https://thirdspacelearning.com/gcse-maths/algebra/power-of-0/
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