Zero Exponent Rule - Examples, Exercises and Solutions

We use the zero exponent rule.

$X^0=1$We obtain

$112^0=1$Therefore, the correct answer is option C.

We use the power property:

$X^0=1$We apply it to the problem:

$5^0=1$Therefore, the correct answer is C.

Let's solve the problem step by step using the Zero Exponent Rule, which states that any non-zero number raised to the power of 0 is equal to 1.

• Consider the expression: $100^0$.
• According to the Zero Exponent Rule, if we have any non-zero number, say $a$, then $a^0 = 1$.
• Here, $a = 100$ which is clearly a non-zero number, so following the rule, we find that:
• $100^0 = 1$.

Therefore, the expression $100^0$ is equivalent to 1.

First we'll use the fact that raising any number to the power of 0 gives the result 1, mathematically:

$X^0=1$

We'll apply this to both the numerator and denominator of the fraction in the problem:

$\frac{4^0\cdot6^7}{36^4\cdot9^0}=\frac{1\cdot6^7}{36^4\cdot1}=\frac{6^7}{36^4}$

Next we'll note that -36 is a power of the number 6:

$36=6^2$

And we'll use this fact in the denominator to get expressions with identical bases in both the numerator and denominator:

$\frac{6^7}{36^4}=\frac{6^7}{(6^2)^4}$

Now we'll recall the power rule for power of a power to simplify the expression in the denominator:

$(a^m)^n=a^{m\cdot n}$

And we'll also recall the power rule for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

We'll apply these two rules to the expression we got above:

$\frac{6^7}{(6^2)^4}=\frac{6^7}{6^{2\cdot4}}=\frac{6^7}{6^8}=6^{7-8}=6^{-1}$

Where in the first stage we applied the first rule we mentioned earlier - the power of a power rule and simplified the expression in the exponent of the denominator term, then in the next stage we applied the second power rule mentioned before - the division rule for terms with identical bases, and again simplified the expression in the resulting exponent,

Finally we'll use the power rule for negative exponents:

$a^{-n}=\frac{1}{a^n}$

And we'll apply it to the expression we got:

$6^{-1}=\frac{1}{6}$

Let's summarize everything we did, we got that:

$\frac{4^0\cdot6^7}{36^4\cdot9^0}=\frac{1}{6}$

Therefore the correct answer is A.

We use the zero exponent rule.

$X^0=1$We obtain:

$\big( \frac{7}{125}\big)^0=1$Therefore, the correct answer is option B.

Due to the fact that raising any number (except zero) to the power of zero will yield the result 1:

$X^0=1$It is thus clear that:

$(\frac{7}{4})^0=1$Therefore, the correct answer is option C.

We use the formula:

$a^0=1$

$\frac{9\times3}{8^0}=\frac{9\times3}{1}=9\times3$

We know that:

$9=3^2$

Therefore, we obtain:

$3^2\times3=3^2\times3^1$

We use the formula:

$a^m\times a^n=a^{m+n}$

$3^2\times3^1=3^{2+1}=3^3$

Due to the fact that raising any number (except zero) to the power of zero will give the result 1:

$X^0=1$Let's examine the expression of the problem:

$(300\cdot\frac{5}{3}\cdot\frac{2}{7})^0$The expression inside of the parentheses is clearly not 0 (it can be calculated numerically and verified)

Therefore, the result of raising to the power of zero will give the result 1, that is:

$(300\cdot\frac{5}{3}\cdot\frac{2}{7})^0 =1$Therefore, the correct answer is option A.

We'll use the law of exponents for negative exponents, but in the opposite direction:

$\frac{1}{a^n} =a^{-n}$Let's apply this law to the problem:

$4^5-4^6\cdot\frac{1}{4}= 4^5-4^6\cdot4^{-1}$When we apply the above law to the second term from the left in the sum, and convert the fraction to a term with a negative exponent,

Next, we'll use the law of exponents for multiplying terms with identical bases:

$a^m\cdot a^n=a^{m+n}$Let's apply this law to the expression we got in the last step:

$4^5-4^6\cdot4^{-1} =4^5-4^{6+(-1)}=4^5-4^{6-1}=4^5-4^{5}=0$When we apply the above law of exponents to the second term from the left in the expression we got in the last step, then we'll simplify the resulting expression,

Let's summarize the solution steps:

$4^5-4^6\cdot\frac{1}{4}= 4^5-4^6\cdot4^{-1} =4^5-4^{5}=0$

We got that the answer is 0,

Therefore the correct answer is answer A.

We'll use the power rule for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$and we'll simplify the second term on the left in the equation using it:
$5^3+5^{-3}\cdot5^3=5^3+5^{-3+3}=5^3+5^0=5^3+1$ where in the first stage we applied the mentioned rule to the second term on the left, then we simplified the expression with the exponent, and in the final stage we used the fact that any number raised to the power of 0 equals 1,

We didn't touch the first term of course since it was already simplified,

Therefore the correct answer is answer C.

We use the formula:

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

We decompose the fraction inside of the parentheses:

$(\frac{1}{7})^4=\frac{1^4}{7^4}$

We obtain:

$7^4\times8^3\times\frac{1^4}{7^4}$

We simplify the powers: $7^4$

We obtain:

$8^3\times1^4$

Remember that the number 1 in any power is equal to 1, thus we obtain:

$8^3\times1=8^3$

We use the law of exponents to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply the law to given the problem:

$7^x\cdot7^{-x}=7^{x+(-x)}=7^{x-x}=7^0$In the first stage we apply the above power rule and in the following stages we simplify the expression obtained in the exponent,

Subsequently, we use the zero power rule:

$X^0=1$We obtain:

$7^0=1$Lastly we summarize the solution to the problem.

$7^x\cdot7^{-x}=7^{x-x}=7^0 =1$Therefore, the correct answer is option B.

First, let's recall the law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

and we'll use it to handle the fraction's denominator in the problem:

$\frac{(-3)^5\cdot8^4}{(-3)^3(-3)^2(-3)^{-5}}=\frac{(-3)^5\cdot8^4}{(-3)^{3+2+(-5)}}=\frac{(-3)^5\cdot8^4}{(-3)^0}$

where in the first stage we'll apply the above law to the denominator and then simplify the expression with the exponent in the denominator,

Now let's remember that raising any number to the power of 0 gives the result 1, or mathematically:

$X^0=1$

therefore the denominator we got in the last stage is 1,

meaning we got that:

$\frac{(-3)^5\cdot8^4}{(-3)^3(-3)^2(-3)^{-5}}=\frac{(-3)^5\cdot8^4}{(-3)^0}=\frac{(-3)^5\cdot8^4}{1}=(-3)^5\cdot8^4$

Now let's recall the law of exponents for an exponent of a product in parentheses:

$(x\cdot y)^n=x^n\cdot y^n$

and we'll apply this law to the first term in the product we got:

$(-3)^5\cdot8^4=(-1\cdot3)^5\cdot8^4 =(-1)^5\cdot3^5\cdot8^4=-1\cdot 3^5\cdot 8^4=-3^5\cdot8^4$

Note that the exponent applies separately to both the number 3 and its sign, which is the minus sign that is actually multiplication by $-1$

Let's summarize everything we did, we got that:

$\frac{(-3)^5\cdot8^4}{(-3)^3(-3)^2(-3)^{-5}}=(-3)^5\cdot8^4 = -3^5\cdot8^4$

Therefore the correct answer is answer C.

Let's start by simplifying the second term in the complete multiplication, meaning - the fraction. We'll simplify it in two stages:

In the first stage we'll use the power law for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

and simplify the fraction's numerator:

$\frac{19^{35}\cdot19^{-32}}{19^3}=\frac{19^{35+(-32)}}{19^3}=\frac{19^{35-32}}{19^3}=\frac{19^3}{19^3}$

Next, we can either remember that dividing any number by itself gives 1, or use the power law for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$ to get that:$\frac{19^3}{19^3}=19^{3-3}=19^0=1$

where in the last step we used the fact that raising any number to the power of 0 gives 1, meaning mathematically that:

$X^0=1$

Let's summarize this part, we got that:

$\frac{19^{35}\cdot19^{-32}}{19^3}=1$

Let's now return to the complete expression in the problem and substitute this result in place of the fraction:

$3^{-3}\cdot\frac{19^{35}\cdot19^{-32}}{19^3}=3^{-3}\cdot1=3^{-3}$

In the next stage we'll recall the power law for negative exponents:

$a^{-n}=\frac{1}{a^n}$

and apply this law to the result we got:

$3^{-3}=\frac{1}{3^3}=\frac{1}{27}$

Summarizing all the steps above, we got that:

$3^{-3}\cdot\frac{19^{35}\cdot19^{-32}}{19^3}=3^{-3}=\frac{1}{27}$

Therefore the correct answer is answer A.

This problem can be solved using the Law of exponents power rules for a negative power, power over a power, as well as the power rule for the product between terms with identical bases.

However we prefer to solve it in a quicker way:

To this end, the power by power law is applied to the parentheses in which the terms are multiplied, but in the opposite direction:

$x^n\cdot y^n=(x\cdot y)^n$Since in the expression in the problem there is a multiplication between two terms with identical powers, this law can be used in its opposite sense.

$5^4\cdot(\frac{1}{5})^4=\big(5\cdot\frac{1}{5}\big)^4$Since the multiplication in the given problem is between terms with the same power, we can apply this law in the opposite direction and write the expression as the multiplication of the bases of the terms in parentheses to which the same power is applied.

We continue and simplify the expression inside of the parentheses. We can do it quickly if inside the parentheses there is a multiplication between two opposite numbers, then their product will give the result: 1, All of the above is applied to the problem leading us to the last step:

$\big(5\cdot\frac{1}{5}\big)^4 = 1^4=1$We remember that raising the number 1 to any power will always give the result: 1, which means that:

$1^x=1$Summarizing the steps to solve the problem, we obtain the following:

$5^4\cdot(\frac{1}{5})^4=\big(5\cdot\frac{1}{5}\big)^4 =1$Therefore, the correct answer is option b.