Solution: Step 1: Denominator Reduction Step 1.1: Calculate 613 Compute the power of 6 raised to 13. frac1613 cdot 816 = frac113060694016 cdot 816 Step 1.2: Calculate 816 Evaluate the power of 8 raised to 16. frac113060694016 cdot 816 = frac113060694016 cdot 281474976710656 Step 2: Denominator Multiplication Multiply the two large numbers to find the combined denominator. frac113060694016 cdot 281474976710656 = frac13676258543978604182634496 Step 3: Final Representation Express the simplified fraction in various formats. - Exact Form: frac13676258543978604182634496 - Decimal Form: 2.72015688 times 10-25 Solution:"The simplified form of the given expression is frac13676258543978604182634496, which can also be expressed as 2.72015688 times 10-25 in decimal notation." Knowledge Notes: 1. Exponentiation: Raising a number to a power means multiplying the number by itself as many times as the value of the power. For example, an means a times a times ... times a (n times). 2. Simplifying Fractions: To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD). However, in this case, since the numerator is 1, we only simplify the denominator. 3. Multiplying Large Numbers: When dealing with very large numbers, it's often easier to work with powers of 10 or scientific notation to simplify the multiplication process. 4. Scientific Notation: This is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used in science and engineering. The format is m times 10n where m is the mantissa and n is the exponent, indicating the power of 10 by which to multiply the mantissa. 5. Decimal Representation: Converting a fraction to decimal form involves division of the numerator by the denominator. When the numbers are very large or very small, scientific notation is used to represent …

Simplify 1/(6^13*8^16)

The problem is asking to simplify the mathematical expression $1/(6^{13} * 8^{16})$. This expression contains exponents and involves multiplying two numbers, 6 and 8, each raised to a power (13 and 16 respectively). The goal is to manipulate the expression to a more simplified form, possibly by reducing the fraction or expressing the numbers in their prime factorized form to reveal potential simplifications.

$\frac{1}{6^{13} \cdot 8^{16}}$

Answer

Expert–verified

Solution:

Step 1: Denominator Reduction

Step 1.1: Calculate $6^{13}$

Compute the power of $6$ raised to $13$. $\frac{1}{6^{13} \cdot 8^{16}} = \frac{1}{13060694016 \cdot 8^{16}}$

Step 1.2: Calculate $8^{16}$

Evaluate the power of $8$ raised to $16$. $\frac{1}{13060694016 \cdot 8^{16}} = \frac{1}{13060694016 \cdot 281474976710656}$

Step 2: Denominator Multiplication

Multiply the two large numbers to find the combined denominator. $\frac{1}{13060694016 \cdot 281474976710656} = \frac{1}{3676258543978604182634496}$

Step 3: Final Representation

Express the simplified fraction in various formats.

Exact Form: $\frac{1}{3676258543978604182634496}$
Decimal Form: $2.72015688 \times 10^{-25}$

Solution:"The simplified form of the given expression is $\frac{1}{3676258543978604182634496}$, which can also be expressed as $2.72015688 \times 10^{-25}$ in decimal notation."

Knowledge Notes:

Exponentiation: Raising a number to a power means multiplying the number by itself as many times as the value of the power. For example, $a^n$ means $a \times a \times ... \times a$ (n times).
Simplifying Fractions: To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD). However, in this case, since the numerator is 1, we only simplify the denominator.
Multiplying Large Numbers: When dealing with very large numbers, it's often easier to work with powers of 10 or scientific notation to simplify the multiplication process.
Scientific Notation: This is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used in science and engineering. The format is $m \times 10^n$ where $m$ is the mantissa and $n$ is the exponent, indicating the power of 10 by which to multiply the mantissa.
Decimal Representation: Converting a fraction to decimal form involves division of the numerator by the denominator. When the numbers are very large or very small, scientific notation is used to represent the decimal.