63 divided 6^ -2
17 answers
To simplify the expression \( 63 \div 6^{-2} \), we can first rewrite the division in terms of multiplication by the reciprocal:
\[ 63 \div 6^{-2} = 63 \times 6^{2} \]
Next, we calculate \( 6^{2} \):
\[ 6^{2} = 36 \]
Now, we substitute this back into the expression:
\[ 63 \times 36 \]
Now, we can calculate \( 63 \times 36 \):
\[ 63 \times 36 = 2268 \]
Therefore, the simplified expression is:
\[ \boxed{2268} \]
To simplify the expression \( \frac{6^3}{6^{-2}} \), we can use the rule of exponents which states that when you divide two powers with the same base, you subtract their exponents:
\[ \frac{a^m}{a^n} = a^{m-n} \]
In this case, we have:
\[ \frac{6^3}{6^{-2}} = 6^{3 - (-2)} = 6^{3 + 2} = 6^{5} \]
Now, we can calculate \( 6^5 \):
\[ 6^5 = 6 \times 6 \times 6 \times 6 \times 6 \]
Calculating step by step:
Thus, \( 6^5 = 7776 \).
Therefore, the simplified expression is:
\[ \boxed{7776} \]
Any non-zero number raised to the power of zero is equal to 1. Therefore, regardless of the values of \( 54^4 \) and \( 56^5 \), the expression \( (54^4 \cdot 56^5)^0 \) simplifies to:
\[ \boxed{1} \]
To simplify the expression \(\frac{(8^3)^5}{8^{17}}\), we can start with the exponentiation part.
Using the power of a power property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify \((8^3)^5\):
\[ (8^3)^5 = 8^{3 \cdot 5} = 8^{15} \]
Now we can substitute this back into the expression:
\[ \frac{8^{15}}{8^{17}} \]
Next, we apply the quotient rule for exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\):
\[ \frac{8^{15}}{8^{17}} = 8^{15 - 17} = 8^{-2} \]
Thus, the expression simplifies to:
\[ 8^{-2} \]
If needed, we can express \(8^{-2}\) as a fraction:
\[ 8^{-2} = \frac{1}{8^2} = \frac{1}{64} \]
So, an equivalent expression is:
\[ 8^{-2} \quad \text{or} \quad \frac{1}{64} \]
Thus, the simplified expression is:
\[ \boxed{8^{-2}} \quad \text{or} \quad \boxed{\frac{1}{64}} \]
To simplify the expression \((5^3 \cdot 5^6)^3\), we can start by using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\).
First, simplify the expression inside the parentheses:
\[ 5^3 \cdot 5^6 = 5^{3+6} = 5^9 \]
Now substitute this back into the expression:
\[ (5^9)^3 \]
Next, use the power of a power property, which states \((a^m)^n = a^{m \cdot n}\):
\[ (5^9)^3 = 5^{9 \cdot 3} = 5^{27} \]
Thus, the expression simplifies to:
\[ \boxed{5^{27}} \]
To simplify the expression \(5^{-2} \cdot 5^7 \div 5^2\), we can proceed step by step using the properties of exponents.
First, let's rewrite the division as multiplication by the reciprocal:
\[ 5^{-2} \cdot 5^7 \div 5^2 = 5^{-2} \cdot 5^7 \cdot 5^{-2} \]
Now we combine the exponents using the property \(a^m \cdot a^n = a^{m+n}\):
\[ 5^{-2} \cdot 5^7 \cdot 5^{-2} = 5^{-2 + 7 - 2} \]
Let's simplify the exponent:
\[ -2 + 7 - 2 = 3 \]
Thus, we get:
\[ 5^3 \]
So, the simplified expression is:
\[ \boxed{5^3} \]
If desired, we can further calculate \(5^3\):
\[ 5^3 = 125 \]
Thus, the final answer can also be expressed as:
\[ \boxed{125} \]
To simplify the expression \(7^7 \times (7^8)^{-3}\), we can start by simplifying the part \( (7^8)^{-3} \) using the property of exponents that states \((a^m)^{-n} = a^{-m \cdot n}\).
First, simplify \( (7^8)^{-3} \):
\[ (7^8)^{-3} = 7^{8 \cdot (-3)} = 7^{-24} \]
Now, substitute this back into the expression:
\[ 7^7 \times 7^{-24} \]
Next, we apply the property of exponents \(a^m \times a^n = a^{m+n}\):
\[ 7^7 \times 7^{-24} = 7^{7 + (-24)} = 7^{7 - 24} = 7^{-17} \]
Thus, the expression simplifies to:
\[ 7^{-17} \]
If desired, we can also express this as a fraction:
\[ 7^{-17} = \frac{1}{7^{17}} \]
So, an equivalent expression is:
\[ \boxed{7^{-17}} \quad \text{or} \quad \boxed{\frac{1}{7^{17}}} \]
To simplify the expression \(\frac{2^{-2}}{2^{-4}}\), we can use the property of exponents concerning division, which states:
\[ \frac{a^m}{a^n} = a^{m-n} \]
Applying this property to our expression:
\[ \frac{2^{-2}}{2^{-4}} = 2^{-2 - (-4)} = 2^{-2 + 4} = 2^{2} \]
Now we can simplify \(2^2\):
\[ 2^{2} = 4 \]
Thus, the simplified expression is:
\[ \boxed{4} \]
To simplify the expression \(\left(\frac{5^7}{5^4}\right)^3\), we can start by simplifying the fraction inside the parentheses using the property of exponents:
\[ \frac{a^m}{a^n} = a^{m-n} \]
So, we have:
\[ \frac{5^7}{5^4} = 5^{7-4} = 5^3 \]
Now, we substitute this result back into the expression:
\[ (5^3)^3 \]
Next, we apply the power of a power property of exponents:
\[ (a^m)^n = a^{m \cdot n} \]
Thus,
\[ (5^3)^3 = 5^{3 \cdot 3} = 5^9 \]
So the simplified expression is:
\[ \boxed{5^9} \]
